Step by step
The question
Question: work out (45 − 19) mod 6.
(45 − 19) mod 6
Mod 6 means: the remainder after dividing by 6.
mod 6 = the remainder after ÷ 6
1 Each remainder
Divide 45 by 6: 6 goes 7 times, that is 42, and 3 is left over.
45 = 6 × 7 + 3
19 is 6 × 3, plus 1.
19 = 6 × 3 + 1
Only the remainders matter: the groups of 6 can never change what is left over.
the 6s do not change the remainder
Careful!
Every number is swapped for its remainder, not just the first one.
Careful: not 3 − 19 ✗
2 Subtract the remainders
So we only need 3 − 1.
(45 − 19) mod 6 = (3 − 1) mod 6
3 minus 1 is 2.
= 2 mod 6
2 is already from 0 to 5.
= 2
3 Check
With a calculator: 45 minus 19 is 26.
That is 6 times 4, plus 2.
The same.
Right!
45 − 19 = 26 = 6 × 4 + 2
Let's look back.
First: each remainder.
Then: subtract the remainders.
Last: check.
Answer
= 2