Modular arithmetic

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.

What is the units digit for the number 123^(456) ?

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

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

What is the remainder when 2^{164}is divided by 7?

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

Here are many ideas for you to investigate - all linked with the number 2000.