# Math Help - prove congruence

1. ## prove congruence

Prove that if a is an odd integer, then a^2 is congruent to 1(mod8)

2. Originally Posted by mandy123
Prove that if a is an odd integer, then a^2 is congruent to 1(mod8)
$a=8k\pm 1,8k\pm 3$.
Now square them.

3. so when I square them how exactly is that going to help me prove that a is congruent to 1(mod8)?

Am I just suppose to give examples
like if k=3
then it would be
625 which is congruent to 1(mod8) and so on, or do I need to continue the proof?

4. Originally Posted by mandy123
so when I square them how exactly is that going to help me prove that a is congruent to 1(mod8)?
$(8k\pm 1)^2 = 8(8k^2) + 8(\pm 2k) + 1 = 8(8k^2 \pm 2k)+1$ thus it is congruent to 1 mod 8.

$(8k\pm 3)^2 = 8(8k^2) + 8(\pm 6k) + 9 = 8(8k^2 \pm 6k + 1)+1$ thus it is congruent to 1 mod 8.

5. oh, ok, i see now how you did it. I was just squaring them completely differently than you. Thank you so much!