Free lab

Modular Arithmetic Lab

Pick a modulus: put integers on the dial and step around it, then open the addition, multiplication and power tables and look for patterns.

Remainder dial

The dial has m slots. Any integer placed on it lands in the slot of its remainder on division by m. Choose “+ s each step” or “× s each step” and walk step by step: do your footprints reach every slot?

Try this: with m = 12, which values of s visit all 12 slots using “+ s”? How are they related to 12? Switch to “× s” and start from 1: with m = 7 and s = 3, how many slots can you reach? And with m = 8?

Remainder tables

Three tables; change the modulus as you like. When does a 0 appear “inside” the multiplication table? In the n² column of the power table, which remainders never appear?

Try this: set the modulus to 8 and look at the n² column of the power table — what remainders can a square leave on division by 8? Then try 3, 4, 5, 9 and 16 and note each one. These “impossible remainders” are the most common weapon for proving that an equation has no integer solutions.

Pick a modulus: is the equation still alive?

If the remainders the left side can take never meet the remainder of the right side, the equation has no integer solutions.