Step by step
The question
Question: work out (123 × 456) mod 7.
(123 × 456) mod 7
Mod 7 means: the remainder after dividing by 7.
mod 7 = the remainder after ÷ 7
1 Each remainder
Divide 123 by 7: 7 goes 17 times, that is 119, and 4 is left over.
123 = 7 × 17 + 4
456 is 7 × 65, plus 1.
456 = 7 × 65 + 1
Only the remainders matter: the groups of 7 can never change what is left over.
the 7s do not change the remainder
Careful!
Every number is swapped for its remainder, not just the first one.
Careful: not 4 × 456 ✗
2 Multiply the remainders
So we only need 4 × 1.
(123 × 456) mod 7 = (4 × 1) mod 7
4 times 1 is 4.
= 4 mod 7
4 is already from 0 to 6.
= 4
3 Check
With a calculator: 123 times 456 is 56088.
That is 7 times 8012, plus 4.
The same.
Right!
123 × 456 = 56088 = 7 × 8012 + 4
Let's look back.
First: each remainder.
Then: multiply the remainders.
Last: check.
Answer
= 4