Is -4 ≡ 11 (mod 5)

Step by step

The question

Question: is −4 ≡ 11 (mod 5)?

This sign means: they leave the same remainder when we divide by 5.

We say congruent.

1 Each remainder

Divide −4 by 5.

For a negative number we go down to the multiple of 5 just below it: 5 × (−1) = −5.

Then up 1.

11 is 5 × 2, plus 1.

Careful!

−4 and 11 are different numbers.

The question is only about their remainders.

2 Compare the remainders

Both leave 1.

The same remainder.

So yes: −4 ≡ 11 (mod 5).

3 Check

Another way: the difference −15 is a multiple of 5.

Same remainders always give a difference that 5 goes into.

Right!

Let's look back.

First: each remainder.

Then: compare the remainders.

Last: check.