Two composite natural numbers n and m are not coprime, and at least one of them is odd.
Recall that the greatest common divisor function,
gcd( , ) returns the largest positive integer
which evenly divides both arguments of the function.
Whilst more than one answer may be true, select the best answer.
The set of (positive) divisors of 6 is
{ 1, 2, 3, 6 }.
The table shows examples of how to count the number of divisors in a given number.
Which of the numbers below has the greatest number of (positive) divisors?
The image shows a BIG calculation. But is the answer correct?
Is the last digit correct? Is the first digit correct? Are the number of digits correct?
An ancient method, more than 1000 years old, is called "casting out the nines".
Sum the digits in the first number. If the result is more than 9, sum the digits again, and repeat. Result A.
Sum the digits in the second number. If the result is more than 9, sum the digits again, and repeat. Result B
Multiply A by B, sum the digits while the result is greater than 9 (as above).
Sum the digits in the product. If the result is more than 9, sum the digits again, and repeat. Result C.
The sum of digits of A times B (reduced as above) must equal C for the result to be possibly correct.
There is a shortcut. If, when summing the digits, you need to sum a 9, then don't bother adding it (cast it away).
In the digit sum, if you encounter digit pairs like 1+8, 2+7, 3+6, 4+5, they add up to 9, so cast them away as well!
A counting number greater than 7 has a remainder of one when divided by 3.
What will the remainder be when the number is multiplied by itself 7 times?
Which is the correct numerical ordering for these fractions, such that the smallest fraction is on the left?
Define n as a natural number > 3
Define q(n) as the product of all primes ≤ n
Define c(n) as the product of all composites ≤ n
For example, q(4) = 2 × 3 = 6 and c(4) = 4.
What is the maximum value of q(n) / c(n)
where we consider the ratio as a real (double precision) value?
How many of the squared 2026 terms are required under the square root symbol in order to make this equality true?
Consider the pair of consecutive natural numbers n and n+1.
What proportion of all these pairs are coprime, which is to say what proportion of these pairs share no common factors (other than one)?
You are familiar with the natural numbers, but maybe they were called counting numbers or whole numbers at the time.
You are familiar with natural numbers in base-10, so that 32 is thirty two, meaning 3 lots of 10 plus 2 units.
What proportion of all natural numbers (expressed in base-10) have all their digits repeated?
For example 33333 is one such special number.
by Leslie Green
A set Lm is defined as the set of natural numbers (positive whole numbers) such that their base-m logarithm is also a natural number.
Which set is bigger (has more elements)?
by Leslie Green
The set L is defined as the set of natural numbers (positive whole numbers) such that their base-10 logarithm is also a natural number.
Which of these is a proper subset of L?
by Leslie Green
We consider the squares of natural numbers.
For example 16 = 42 is such a value.
What proportion (fraction) of perfect squares of naturals are divisible by 9?
by Leslie Green
Multiplicative magic square
Write different counting numbers into small squares of the 3x3 large square so that products of the numbers in each row, each column, and both main diagonals are the same.
Try to find the smallest possible numbers.
Which is the largest of the 9 numbers?
The number 13 is unlucky for many western people.
How many natural numbers are there that are 13 times greater than the sum of its digits?
"Recreational math problems", 1995 by A.P. Savin
Evaluate the infinite product shown.
HINT: Consider the difference of squares, and telescoping products.
We start from the set of natural numbers, which is
{ 1, 2, 3, 4, 5, 6, 7, ... }.
Using the pattern: "discard, discard, keep, discard, discard, keep,..." on consecutive elements of the set of natural numbers we form a new set.
On this new set we then use the pattern: "keep, discard, keep, discard, ...", again starting from the beginning of the set.
What is the form of this final set, given that n is a natural number?
Examples of twin primes are
{3, 5},
{5, 7},
{11, 13},
{17, 19}.
Our proof shows that
there are infinitely many of these pairs.
At what step does the proof fail?
Since ancient times, people have been interested in equations involving whole numbers. This is one such equation.
When k = 1, how many solutions does this equation have?
The upper set in each pair is the set of natural numbers, represented by the double-struck N. The three dots (ellipsis) shows that each set continues in the same manner without limit.
In the image, the four sets ( A, B, C, and D ) are all subsets of the naturals. The subsets are all matched with the naturals in particular patterns, but not all of the patterns are genuine, the rest being invalid. What makes the pattern valid? There is a particular symmetry required, known as translation invariance. You may remember translation from geometry lessons where translation is a sliding (or shifting) without rotation and without scaling.
We can't explicitly write down the pattern all the way to infinity. We therefore have to imagine what happens, and see if the pattern continues without change as we move down the number line. Which matching patterns show the required symmetry?
by Leslie Green
A particular function (with a natural argument) returns the sum of the squared digits of the argument when represented as a base 10 number.
For example when given 121 the function would return 12 + 22 + 12 = 6.
What do you expect to happen if the function output is used as its input, over and over again, starting from any natural value?
If we were to suppose that there is a largest natural number, how would this compare to the supposed largest integer?
The image shows Galilean 1:1 correspondence between the natural numbers (with 0 included) and the integers.
Be careful of the symbol for the set of natural numbers, it is a double-struck capital N. If the N has a 'bendy body' it may be the symbol aleph, which means something else entirely.
by Leslie Green
Which infinite set has more elements?
HINT: Do not compare the infinite sets using Galilean 1:1 correspondence as that obsolete method tells you nothing about the size of infinite sets.
by Leslie Green
Amongst the natural numbers, what is the probability of randomly picking a value which is the sum of exactly one element from set A and one element from set B?
For any two consecutive natural numbers, exactly 1 is divisible by 2.
For any three consecutive natural numbers, exactly 1 is divisible by 3.
For any six consecutive natural numbers, how many are exactly divisible by either 2 or 3?
by Leslie Green
The image shows a table of all possible rational numbers when using only the first 6 natural numbers.
What fraction of all rational numbers are natural numbers?
(HINT: consider the first H natural numbers, in the limit as H tends to infinity)
by Leslie Green
The even natural numbers form an infinite set:
{evens} = { 2, 4, 6, 8, 10, 12, 14, 16, ... }
The odd natural numbers also form an infinite set:
{odds} = {1, 3, 5, 7, 9, 11, 13, 15, ... }
These are disjoint sets, which means they share no common elements. Thus there are no odd numbers in the even-numbers set, and likewise no even numbers in the odd-numbers set.
When we take the union of the odds and evens there are no duplicates (since the sets are disjoint), and the combined set consists of all the natural numbers.
The probability of randomly picking an odd value from the natural numbers is 1/2. The probability of randomly picking an even value from the natural numbers is also 1/2. We say this is evidence that there are as many odd natural numbers as even natural numbers, regardless of any concerns about infinity.
We further say that there are twice as many natural numbers as even natural numbers, since the probability of finding an even natural number within the overall set of natural numbers is 1/2.
Comment on the proposition.
by Leslie Green
I claim that the probability of randomly selecting an even value from non-negative integers up to an indefinitely large value is 50%.
Do you agree?
by Leslie Green
If you were asked to evaluate 3 + 4 = ? you would immediately give the answer 7. You know that the plus-sign means to add the two numbers either side of the plus sign. In advanced work we would call the plus sign a binary operator, where binary refers to the two numbers, and is nothing to do with the binary number system. You are familiar with the binary operators +, - , x, / from your early school work.
In the image we have defined a new binary operator using a fairly unfamiliar symbol (unless you happen to read/write in Greek).
Your task is to evaluate the expression given in (e). This is far from easy. Take as long as you like.
The answer itself is fairly uninteresting. The interesting part is trying to work out how to find the answer from the near infinite number of possibilities.
Find the smallest natural number that is divisible by any natural number from 1 to 25 inclusive.
Remember: 4! = 4 x 3 x 2 x 1
For non-negative integer A, is the product of the three factors A plus one, A plus two, and A plus three always divisible by 6?
Given that N is a positive integer, is one plus the product of the first N consecutive prime numbers itself prime?
Real number r is equal to the pth root of n over m, where both n and m are coprime natural numbers.
To be clear, if p = 2 that is a square root, and if p = 3 that is a cube root.
p is a natural number greater than 1.
State the necessary and sufficient condition for r to be rational.
To be clear, for 9 = 32, the index (power) of the factor 3 is 2.
by Leslie Green
What is the probability that three randomly chosen consecutive natural numbers are pairwise coprime?
Being pairwise coprime means that for the three natural numbers a, b, and c, each pair is coprime, meaning a and b share no common factors (other than 1), likewise for a and c, and likewise for b and c.
by Leslie Green
The probability that two randomly chosen natural numbers are coprime (they share no common factor, other than 1) is
6/π2 ≅ 0.607927.
What is the probability that two randomly chosen consecutive natural numbers are coprime?
The natural numbers are { 1, 2, 3, 4, 5, 6, 7, 8, ... }
by Leslie Green
If n and m are natural numbers (positive integers) then their ratio, q, is a positive rational number.
If n and m are chosen randomly, the probability that q in its most reduced form is 6 / π2 ≅ 0.60793.
q is in its most reduced form when n and m are coprime, meaning they share no common factors.
Suppose that m is some specific prime, p.
What is the probability that q is then in its most reduced form?
The image shows a grid of natural numbers, n and m. Rays emanate from the origin (0,0) to all 'visible' grid points (lattice points).
Like any true geometric point, grid points have no dimension (size), only position. Likewise the rays only have a length dimension.
A grid point is only 'visible' from the origin if there is no earlier grid point in the way. The ray terminates as soon as it hits the first grid point on its path.
The diagram does not correctly illustrate the termination of rays, since some rays pass through multiple grid points.
What necessary condition on n and m determines the visibility of grid points?
by Leslie Green
Consider any general irrational number j, and any non-zero general rational number q.
Which rule creates a new irrational number?
by Leslie Green
The image shows several types of number sets in the form of a Venn diagram.
What is the biggest problem with the diagram?
by Leslie Green
If you randomly picked a natural number greater than say 100, is it more likely to be a perfect square than a perfect cube?
If n is a natural number then a number such as n3 is a perfect cube, whereas a number of the form n2 is a perfect square.
Tracy claims that for any natural number (aka positive integer), n, which you pick, she can find other natural numbers a and b to satisfy the given equality.
Pick the value of n at and above which Tracy is correct.
So, for example, if you pick an answer of 5 you are saying that {5, 6, 7, 8, 9, ... } can all be formed, without gaps.
Suppose that you deterministically choose natural numbers (counting numbers) not greater than R, where R > 9.
Further suppose that you then take each such number, and evaluate its modulus to the fixed natural value N, with N = R2.
How many numbers can be chosen such that no two of them have the same mod N value?
Note that 0 is not considered as a natural number.
If you were to randomly pick whole numbers in the range around 1 billion, which would you be more likely to find?
A number congruent with 5 mod 11 could be written as (11k + 5) for non-negative integer k.
by Leslie Green
Considering natural numbers, starting from 1, what proportion are divisible by any of 2, 3, 5, or 7?
WARNING! This is a hard problem, which is worth spending some time over. Skip it for now if you are in a hurry.
by Leslie Green
Let's read the image information out aloud, assuming that you are blind (to mathematical notation).
"For r, which is an element of the set of positive real numbers, with r being strictly greater than one, and with n being a natural number, it is asserted as true that r raised to the power n is strictly greater than 2, provided that n is sufficiently large".
By way of explanation, we can then tell you that real numbers are any numbers which can be expressed in decimal form. For example 3.1415926 is a real number. Natural numbers are just the counting numbers: 1, 2, 3, 4 ...
Comment on the assertion made in the image.
(HINT: consider the binomial expansion.)
by Leslie Green
Consider the statement shown in the image, and comment on its accuracy.
The natural numbers are simply the counting numbers: 1, 2, 3, 4 …
Some people consider 0 to be included in the natural numbers, and others don't. Here we neglect 0.
Starting from the blue shaded cell with 6 in it, can you find a route only touching blue squares, from one blue square to the next, with each next square's value being two more than the previous square? The route must consist of a rule (or set of rules) which takes you to the next squares, one after another.
The rule can be arbitrarily complicated, for example:
"Go right by N mod 3 squares (where N is the value of the current square)
then go down by N mod 7 squares if N mod 5 is 3, otherwise ... "
It is not obvious how to factor (factorise) the expression on the right. You basically have to try different multiplications and see which ones work. Here we are interested in the general pattern, rather than the specific case.
Having seen the general pattern, we then need to apply that to a special case.
Find a factor of (228 + 1).
by Leslie Green
Divisibility by 3 can be checked by summing the digits of a number and seeing if that sum is itself divisible by 3. If it is, then the number is also divisible by 3.
Here we ask about the divisibility of this small expression, so the rules are quite different. We already know that (22n - 1) is divisible by 3. But does that help us to decide if some other number, (22n+1 + 1), is divisible by 3?
There is a remarkably simple proof, but you do have to know how to use indices correctly!
by Leslie Green
Can you solve the riddle inscribed on this stone tablet, apparently carved by the Ancients?
The almost equally ancient history professor has suggested a translation, but he can't do maths, so he has no clue what the answer to the riddle is.
TRANSLATION:
"Is it true that any number of the form (22n + 1) is a multiple of 3, where n and k are natural (whole) numbers?
We are told that 3 is a factor of (22n - 1)."
Let's read the stuff in the image so it is well understood.
It is a true statement that 9 evenly divides (26 - 1). This is equivalent to saying that (26 - 1) is equal to 9 times some integer constant k, where k is defined to be an element (member) of the set of natural numbers (counting numbers).
We ask, is it true that (26n - 1) is also divisible by 9?
(k is a 'dummy value', so it is not the same for all values of n)
by Leslie Green
Here we consider only natural numbers, also known as counting numbers, whole numbers, and positive integers.
Consider an infinite set of natural numbers generated by some sort of rule of your choice. For example the n-th element in the set might be k raised to the power of n. Call this first infinite set F.
Consider now a second infinite set, also generated by some rule of your choice. Call this second infinite set S.
By suitable choices, is it possible that the sets F and S are disjoint, by which is meant that no element in set F occurs in set S?
Pick the most inclusive correct answer.
by Leslie Green
Take a large (whole) number N and partition it into a Lower part, L, having exactly two digits, and an Upper part, U, having the remaining digits.
Form a new number N' = 2U + L.
It is claimed that if N is divisible by 7, then N' is also divisible by 7.
Comment on this claim.
by Leslie Green
If you can't read the image, it is simply that for one or more of the symbols you do not know their definition.
Here we will read the question to you so that you at least understand what is being asked of you.
"For all n, such that n is an element (member) of the set of natural numbers (a positive whole number) is there a value (assumed to be a whole number) which (evenly) divides the expression in brackets?"
by Leslie Green
The expression on the left is known as a Fermat Number, provided n is a natural number (counting number).
Fermat proposed this formula (in 1640) thinking the values would be prime, and they were called Fermat Primes at one time.
Here we ask, is any such Fermat Number evenly divisible by 3?
by Leslie Green
We assert as being self-evidently true that H + 1 can be simplified to H without significant loss of precision, provided H is sufficiently large. Imagine, for example, that H is an integer with 1000 decimal digits.
The question we now ask is:
"Can we make a similar simplification with the base-2 logarithm of H?"
by Leslie Green
A Farey sequence is a collection of fractional values in their most reduced form, and in the inclusive range between 0 and 1.
Here we show a third-order Farey sequence, which has denominators (downstairs) up to and including three. We consider only the numerical gaps between the adjacent pairs of terms when sorted into ascending order (as we have done here).
Which gap(s) is/are the largest?
The question suggested by the image concerns only positive integer values of n and k.
Is 22n - 1 (evenly) divisible by 3?
We could have asked about a modulus 3 residue of 0, but we asked if it equals an integer multiple of 3 just for variety.
by Leslie Green
The sum and the product of five positive integer numbers are equal.
What is the maximum possible value of any of the numbers?
A perfect square is an integer formed as the square of another integer.
The percentage of all the counting numbers which are perfect squares could be written as 0.000...000% where the three dots (ellipsis) suggests that an arbitrary number of zeros could be placed there.
What should we not say about this situation?
by Leslie Green
We suppose that there is some limiting integer value, V, such that the percentage of perfect squares up to V is less than 0.000% when expressed to three significant digits of percentage, as we have done.
What is the value of V?
Considering all possible whole numbers, what fraction of them are evenly divisible by 3 or 4, or both.
Using the digits 0 to 9 as often as you like, form two 5 digit numbers such that the digits in any particular position for both numbers are not equal. In other words the units digits of each number are different to each other, the tens digits are different to each other, and so on.
What is the minimum possible difference between these two numbers?
James chooses 5 consecutive positive integers so that the sum of two of them is equal to the sum of the three others.
What is the smallest possible value of the sum?
Estimate the sum of all four-digit numbers which consist of the digits 1, 2, 3, and 4 in some order, with each digit appearing exactly once.
The product of 4 strictly positive unique rational numbers is 7. Given that at least three of these numbers are also integers, and 7 is the greatest value of the four, how many different solutions exist?
(Reminder: A rational number is the ratio of two integers, so 4 is an integer -- but could be expressed as the rational value 4/1.)
by Leslie Green
The square root of two is irrational. This does not mean it is a bit crazy, it just means that it cannot be completely represented by the ratio of two integers. It cannot be fully represented by a finite-precision decimal expression, and has a digit sequence which does not repeat.
We assert as fact that any product of a rational number and an irrational number is still irrational.
However, if we multiply √2 by another irrational value, what can we not say?
(Pick the most incorrect statement)
by Leslie Green
The inner circle has a radius equal to the side of the square. The outer circle has a radius equal to the diagonal of the square. The ratio of circumferences is therefore √2, which is irrational.
Two radial lines cross both circles, the crossing points being marked by small blobs. Suppose the angle between the radial lines decreases until the two blobs on the outer circle are infinitely close to each other, and the blobs represent the smallest possible indivisible element of the circle. But the two blobs on the inner circle are closer together, and also indivisible. Additionally, if there are an integer number of indivisible blobs in the outer circle, given that the ratio of blobs is irrational, how can the blobs on the inner circle be a whole number of indivisible elements? (Such reasoning is due to Aristotle from the 4th century BCE.)
Is it reasonable to conclude that matter cannot be composed of indivisible elements?
(Be prepared to argue your case.)
by Leslie Green
The image shows the systematic arrangement of rational numbers proposed by Georg Cantor (1845-1918), but limited to those formed from the first 10 counting numbers.
If instead of 10 counting numbers we use H, where H is both Huge and even, which rational number is closest to but less than 2/H?
by Leslie Green
Each circle contains a unique non-zero decimal digit. The sums of the digits along any of the straight red lines shown are all the same.
To be clear, each sum contains the digits from exactly three circles.
Which digit is found in the middle circle?
by Leslie Green
We have two unique integers, each in the range 2-100 inclusive, which we could arbitrarily label as A and B. In any combination of these numbers we use at most 1 of each number.
We wish to count how many numerically distinct results there could be when we (optionally) combine the numbers using only the (binary) operators plus, minus, multiply, and divide.
Note: (A + B) is clearly never distinct from (B + A).
Note: Results such as (-A) are not considered as the minus sign is being used as a unary operator.
How many numerically distinct results could there be?
by Leslie Green
'Smooth numbers' are those that factor easily in the sense that all their prime factors are lower than some specified limit. If we take a block of consecutive integers, it is obvious that half will be divisible by 2, one third will be divisible by 3, and so on. But the question "How many will not have a prime factor above the square root of the number being tested?" is somewhat more difficult. This graph answers such questions.
If you consider integers in the region of 1E8 (100 million), around 27% of them will not have a prime factor larger than 1E4 (10,000).
The question for you is this: If you consider random integers around 1E18, what percentage of them will not have a prime factor exceeding 1,000,000?
by Leslie Green
You wish to make a table of all the prime factors of all the numbers between 1 and 1,000,000.
For example the number 12 would be written as 12 = 22 x 3.
and you would count 12 as having 3 (not necessarily unique) prime factors.
How many prime factors should you allow in your table?
by Leslie Green
What is the least restrictive valid condition for which this congruence is true?
( p is not zero )
(HINT: Maybe try an easier one first? )
by Leslie Green
Bob has N different tools on his workbench. All N could fit in his tool bag to go with him on a job, but he wonders how many different combinations of 1 or more tools could he choose.
(Hint: how many would it be if he only had 3 different tools?)
by Leslie Green
A Chinese general has well trained troops. On the command THREE they form three columns, but in the first row there are only two soldiers. FIVE puts soldiers in 5 columns, but has 4 soldiers in the first row. SEVEN has 6 soldiers in the first row. ELEVEN has 10 soldiers in the first row. In all formations the soldiers are aligned perfectly except in the first row.
How many soldiers does the general have in this brigade?
by Leslie Green
When faced with such a multiplication, it is important to be careful and systematic (methodical).
There are three terms times three terms, so we expect 9 products terms.
Notice how we add the indexes (indices) of the powers of two when they are multiplied.
When we simplify the terms on the right hand side (RHS), how many are left?
by Leslie Green
The statement
3 | 6
is read as "3 divides 6", and means that 3 is a factor of 6.
We are looking for a factor of the stupidly large (92 digit) Mersenne number shown in the image.
by Leslie Green
N mod M
means subtract M from N as many times as possible until the result (the residue) is between 0 and (M-1) inclusive.
Given that:
N mod 12 = 4
What is:
N mod 3 = ?
by Leslie Green
Given that A and B are integers:
If A is divided by B, the remainder is 6.
If 3A is divided by B, the remainder is 4.
What is the value of B?
This is a bit tricky. In the context of modular arithmetic the 3-bar equal sign doesn't mean "identically equal to", it means "is congruent with ... to the modulus stated at the end of the line".
This is very ugly, but standard notation.
8 is congruent with 5 modulus 3.
The next notation is 3 | 9, which is read as "3 (evenly) divides 9". ... There is no left over part.
If a is congruent with b (mod m), which statement is true?
Of the tests suggested below, which gives the best discrimination between prime and composite numbers above 4?
(In other words, which test finds the most primes compared to composites.)
by Leslie Green
Prime numbers are fascinating things which defy the efforts of mathematicians to predict their pattern within the counting numbers.
Computers are able to quickly test numbers to see if any particular number is prime, but what test(s) do we need to apply?
by Leslie Green
Whilst nobody has come up with a way of predicting which numbers will be prime, there is nevertheless considerable regularity in the distribution of primes on average. For example, the mean gap between primes increases in a very regular manner (as shown). Whilst for all primes below 1000 the average gap between primes is around 6, this figure increases to 22 when we consider all primes below 10 billion.
The Twin Primes conjecture says that there are an infinite number of primes with a gap of 2 between them (eg 5 and 7).
On the basis of this mean prime gap graph alone, comment on the twin primes conjecture.
by Leslie Green
The image shows the systematic arrangement of rational numbers, but limited to those formed from only the first 10 counting numbers. Suppose instead we used the first 1,000 counting numbers.
Which value in the table is closest to 1/2, but less than it?
(Don't use a calculator.)
by Leslie Green
A rational number is one which is formed from the ratio of two counting numbers.
Suppose we have counting numbers p, q, r, and s.
We further suppose that p and q are coprime (no common factors). Likewise r and s are coprime.
We then ask this question: Is it possible for p/q to be exactly equal to r/s when q and s are not equal?
by Leslie Green
A rational number is defined as a number of the form N/M, where both N and M are counting numbers (positive integers).
The Big O notation is used to specify how a parameter affects a result. Suppose the time a computer program takes to run is proportional to the number of elements N being considered. This runtime is stated as O(N). If the runtime depends on the square of the number of elements then it would be O(N2).
O(N) means k x N, where k is not specified.
O(N2) means k x N2, where this k is probably not the same as the previous one, and is again unspecified.
O(N2) is not necessarily greater than O(N) for some value of N, but as N increases, O(N2) will eventually become larger than O(N).
For a large value of N, how many unique rational numbers can be created using only the counting numbers from 1 to N?
by Leslie Green
Present 100 as a sum of positive integers such that their product is at its maximum value.
How many numbers do you use?
The numbers
129
CXXIX
10000001
81
all have the same value.
Why is CXXIX the odd one out?
Author: Leslie Green
Shakuntala Devi was undoubtedly the most brilliant arithmetic mental calculator of all time. In 1977 she mentally calculated the 23rd root of a 201 digit number in a mere 50 seconds. She toured the world showing how she could do calculations faster than they could be entered into and solved by the computers of the day.
The problem for you is much simpler: Evaluate (without using a calculator) the 20th root of the 11 digit number consisting of 1 followed by all zeros.
Author: Leslie Green
Leslie Green asks:
What is the next number in the sequence?
0, 3, 1, 4, 2
(NOTE: It is a single sequence and not two sequences interleaved.)
A magic square is an arrangement of integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the diagonals, all add up to the same number.
Place numbers 13, 14, 15, and 16 in the square to make a magic square.
Which number do you place in the top left cell?
The natural numbers are the counting numbers, like 1, 2, 3, et cetera.
An even number is a natural number which is "evenly divisible" by two.
An odd number is a natural number which is not a multiple of two.
Which statement is correct?
The average of 99 consecutive numbers is 99.
What is the average of the nine smallest of these numbers?
What is the sum of the first 100 prime numbers?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Savin's problem:
Using each of the digits 1,2,3, and 4 twice, write out an eight-digit number in which there is one digit between the ones, two digits between the twos, three digits between the threes, and four digits between the fours.
How many such numbers?
Make two fractions with all the digits from 0 to 9.
Use each digit once.
They add up to exactly 1.
What is one of the fractions?
The first person is 100 cm tall.
Each subsequent person is 20% taller than the person before.
What will be the height of the fifth person?
Is the given value evenly divisible by 11? 
n and k are both natural numbers. 
Evaluate the equation given. 
Without solving for the values of each symbol, which symbol has the largest value?
What is the Least Common Multiple (LCM) of the coprime composite numbers m, n, t?
Which definition of the vertical bars shown in the image is incorrect?
Has the infinite series S been summed correctly? 
All these expressions are related. 
The image shows the symbol for infinity. 
Is the number shown prime?
Comment on this theorem and its proof. 
What is the correct value of this strange looking expression?
What is the probability that the number shown is prime? 
0.999 . . . means infinitely repeating decimals. 
I multiply the two page numbers and get 6162. 
With a 1 after a nine-digit number, it is the same as a 1 before it. 
If you wrote all the whole numbers from 1 through 1000, how many times would you write the digit 4?
Which gives the largest answer?
Which gives the largest answer?
Find the hundreds digit (third digit from the right) of the product: 
The last digit of the number 777 is