Step by step
The question
Question: is 29 ≡ 5 (mod 8)?
29 ≡ 5 (mod 8) ?
This sign means: they leave the same remainder when we divide by 8.
We say congruent.
≡ (mod n): the same remainder after ÷ n
1 Each remainder
Divide 29 by 8: 8 goes 3 times, that is 24, and 5 is left over.
29 = 8 × 3 + 5
5 is 8 × 0, plus 5.
5 = 8 × 0 + 5
Careful!
29 and 5 are different numbers.
The question is only about their remainders.
Careful: compare the remainders, not the numbers
2 Compare the remainders
Both leave 5.
The same remainder.
= 5 = 5
So yes: 29 ≡ 5 (mod 8).
= yes: 29 ≡ 5 (mod 8)
3 Check
Another way: the difference 24 is a multiple of 8.
Same remainders always give a difference that 8 goes into.
Right!
29 − 5 = 24 = 8 × 3
Let's look back.
First: each remainder.
Then: compare the remainders.
Last: check.
Answer
= yes: 29 ≡ 5 (mod 8)