Step by step
The question
Question: work out −13 mod 5.
−13 mod 5
Mod 5 means: the remainder after dividing by 5.
It is always a whole number from 0 to 4.
mod 5 = the remainder after ÷ 5
always 0 to 4
1 Divide by 5
Divide −13 by 5.
For a negative number we go down to the multiple of 5 just below it: 5 × (−3) = −15.
Then up 2.
−13 = 5 × (−3) + 2
Careful!
A remainder is never negative.
We go down to the multiple just below, then count up.
Careful: not −13 mod 5 = −3 ✗
2 On the clock
Think of a clock with 5 numbers, 0 to 4.
So −13 mod 5 = 2.
−13 mod 5 = 2
3 Check
Multiply back: 5 times (−3), plus 2, is −13.
5 × (−3) + 2 = −15 + 2 = −13
And 2 is from 0 to 4.
Right!
0 ≤ 2 < 5
Let's look back.
First: divide by 5.
Then: on the clock.
Last: check.
Answer
−13 mod 5 = 2