Step by step
The question
Question: is 45 ≡ 17 (mod 7)?
45 ≡ 17 (mod 7) ?
This sign means: they leave the same remainder when we divide by 7.
We say congruent.
≡ (mod n): the same remainder after ÷ n
1 Each remainder
Divide 45 by 7: 7 goes 6 times, that is 42, and 3 is left over.
45 = 7 × 6 + 3
17 is 7 × 2, plus 3.
17 = 7 × 2 + 3
Careful!
45 and 17 are different numbers.
The question is only about their remainders.
Careful: compare the remainders, not the numbers
2 Compare the remainders
Both leave 3.
The same remainder.
= 3 = 3
So yes: 45 ≡ 17 (mod 7).
= yes: 45 ≡ 17 (mod 7)
3 Check
Another way: the difference 28 is a multiple of 7.
Same remainders always give a difference that 7 goes into.
Right!
45 − 17 = 28 = 7 × 4
Let's look back.
First: each remainder.
Then: compare the remainders.
Last: check.
Answer
= yes: 45 ≡ 17 (mod 7)