Factors and multiples

problem
The public key

Age

16 to 18

Challenge level

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.
problem
A mixed-up clock

Age

7 to 11

Challenge level

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
problem
Polite numbers

Age

16 to 18

Challenge level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?
problem
Multiplication magic

Age

14 to 16

Challenge level

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.
problem
Six in a circle

Age

5 to 7

Challenge level

If there is a ring of six chairs and thirty children must either sit on a chair or stand behind one, how many children will be behind each chair?
problem
Phew I'm factored

Age

14 to 16

Challenge level

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.
problem
Powerful factors

Age

16 to 18

Challenge level

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.
problem
Factorial fun

Age

16 to 18

Challenge level

How many divisors does factorial n (n!) have?
problem
Dirisibly yours

Age

16 to 18

Challenge level

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
problem
A square deal

Age

7 to 11

Challenge level

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.