Modular arithmetic

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Find the remainder when 3^{2001} is divided by 7.

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.

How many different ways can you arrange the officers in a square?

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?

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

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?

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

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

Choose any whole number n, cube it, add 11n, and divide by 6. What do you notice?