Divisibility

problem

Elevens

Age

16 to 18

Challenge level

1 out of 3

Add powers of 3 and powers of 7 and get multiples of 11.
problem

Repeaters

Age

11 to 14

Challenge level

2 out of 3

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
problem

Digat

Age

11 to 14

Challenge level

2 out of 3

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
problem

Divisively So

Age

11 to 14

Challenge level

2 out of 3

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

Remainder

Age

11 to 14

Challenge level

2 out of 3

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

Squaresearch

Age

14 to 16

Challenge level

3 out of 3

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
problem

Eminit

Age

11 to 14

Challenge level

3 out of 3

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?
problem

Just Repeat

Age

11 to 14

Challenge level

3 out of 3

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

Three Times Seven

Age

11 to 14

Challenge level

3 out of 3

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

Book Codes

Age

7 to 11

Challenge level

2 out of 3

Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?