Divisibility

problem
Favourite

Prime AP

Age

16 to 18

Challenge level

1 out of 3

What can you say about the common difference of an AP where every term is prime?
problem

Novemberish

Age

14 to 16

Challenge level

1 out of 3

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.
problem

Check Codes

Age

14 to 16

Challenge level

1 out of 3

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?
problem

Transposition Fix

Age

14 to 16

Challenge level

1 out of 3

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?
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?