Step by step
The question
Question: work out (38 + 27) mod 5.
(38 + 27) mod 5
Mod 5 means: the remainder after dividing by 5.
mod 5 = the remainder after ÷ 5
1 Each remainder
Divide 38 by 5: 5 goes 7 times, that is 35, and 3 is left over.
38 = 5 × 7 + 3
27 is 5 × 5, plus 2.
27 = 5 × 5 + 2
Only the remainders matter: the groups of 5 can never change what is left over.
the 5s do not change the remainder
Careful!
Every number is swapped for its remainder, not just the first one.
Careful: not 3 + 27 ✗
2 Add the remainders
So we only need 3 + 2.
(38 + 27) mod 5 = (3 + 2) mod 5
3 plus 2 is 5.
= 5 mod 5
5 is 5 × 1 + 0.
The remainder is 0.
= 0
5 = 5 × 1 + 0
3 Check
With a calculator: 38 plus 27 is 65.
That is 5 times 13, plus 0.
The same.
Right!
38 + 27 = 65 = 5 × 13 + 0
Let's look back.
First: each remainder.
Then: add the remainders.
Last: check.
Answer
5 = 5 × 1 + 0