Place value

problem
What an odd fact(or)

Age

11 to 14

Challenge level

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?
problem
The Deca Tree

Age

7 to 11

Challenge level

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.
problem
Trebling

Age

7 to 11

Challenge level

Can you replace the letters with numbers? Is there only one solution in each case?
problem
Arrange the digits

Age

11 to 14

Challenge level

Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?
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
Sixty-seven squared

Age

16 to 18

Challenge level

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?
problem
Three times seven

Age

11 to 14

Challenge level

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
problem
Six times five

Age

11 to 14

Challenge level

How many six digit numbers are there which DO NOT contain a 5?
problem
Just repeat

Age

11 to 14

Challenge level

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?
problem
Which is quicker?

Age

7 to 11

Challenge level

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?