Modular arithmetic

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

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.

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?

What are the possible remainders when the 100-th power of an integer is divided by 125?

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

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

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.

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

What remainders do you get when square numbers are divided by 4?