Factors and multiples

problem
Heads and feet

Age

5 to 7

Challenge level

On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there?
problem
Remainder

Age

11 to 14

Challenge level

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
problem
Number rules - OK

Age

14 to 16

Challenge level

Can you produce convincing arguments that a selection of statements about numbers are true?
problem
X marks the spot

Age

11 to 14

Challenge level

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .
problem
Really Mr. Bond

Age

14 to 16

Challenge level

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
problem
Divisively so

Age

11 to 14

Challenge level

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
problem
Power crazy

Age

11 to 14

Challenge level

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
problem
Powerful factorial

Age

11 to 14

Challenge level

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?
problem
Great Granddad

Age

11 to 14

Challenge level

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?
problem
Eminit

Age

11 to 14

Challenge level

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?