Step by step
The question
Question: work out (87 × 95) mod 11.
(87 × 95) mod 11
Mod 11 means: the remainder after dividing by 11.
mod 11 = the remainder after ÷ 11
1 Each remainder
Divide 87 by 11: 11 goes 7 times, that is 77, and 10 is left over.
87 = 11 × 7 + 10
95 is 11 × 8, plus 7.
95 = 11 × 8 + 7
Only the remainders matter: the groups of 11 can never change what is left over.
the 11s do not change the remainder
Careful!
Every number is swapped for its remainder, not just the first one.
Careful: not 10 × 95 ✗
2 Multiply the remainders
So we only need 10 × 7.
(87 × 95) mod 11 = (10 × 7) mod 11
10 times 7 is 70.
= 70 mod 11
70 is 11 × 6 + 4.
The remainder is 4.
= 4
70 = 11 × 6 + 4
3 Check
With a calculator: 87 times 95 is 8265.
That is 11 times 751, plus 4.
The same.
Right!
87 × 95 = 8265 = 11 × 751 + 4
Let's look back.
First: each remainder.
Then: multiply the remainders.
Last: check.
Answer
70 = 11 × 6 + 4