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 Multiply the remainders
So we only need 3 × 2.
(38 × 27) mod 5 = (3 × 2) mod 5
3 times 2 is 6.
= 6 mod 5
6 is 5 × 1 + 1.
The remainder is 1.
= 1
6 = 5 × 1 + 1
3 Check
With a calculator: 38 times 27 is 1026.
That is 5 times 205, plus 1.
The same.
Right!
38 × 27 = 1026 = 5 × 205 + 1
Let's look back.
First: each remainder.
Then: multiply the remainders.
Last: check.
Answer
6 = 5 × 1 + 1