Step by step
The question
Question: is 38 ≡ 14 (mod 6)?
38 ≡ 14 (mod 6) ?
This sign means: they leave the same remainder when we divide by 6.
We say congruent.
≡ (mod n): the same remainder after ÷ n
1 Each remainder
Divide 38 by 6: 6 goes 6 times, that is 36, and 2 is left over.
38 = 6 × 6 + 2
14 is 6 × 2, plus 2.
14 = 6 × 2 + 2
Careful!
38 and 14 are different numbers.
The question is only about their remainders.
Careful: compare the remainders, not the numbers
2 Compare the remainders
Both leave 2.
The same remainder.
= 2 = 2
So yes: 38 ≡ 14 (mod 6).
= yes: 38 ≡ 14 (mod 6)
3 Check
Another way: the difference 24 is a multiple of 6.
Same remainders always give a difference that 6 goes into.
Right!
38 − 14 = 24 = 6 × 4
Let's look back.
First: each remainder.
Then: compare the remainders.
Last: check.
Answer
= yes: 38 ≡ 14 (mod 6)