1 Show that every square is congruent to 0 1 or 1 modulo 5 a

1. Show that every square is congruent to 0, 1 or -1 modulo 5, and is congruent to 0 ,1 or 4 modulo 8.

2. Let p be a prime number and k a positive integer.

a) Show that if x is an integer such that x^2 x mod p, then x 0 or 1 mod p.

b) Show that if x is an integer such that x^2 x mod p^k, then x 0 or 1 mod p^k.

I need help understanding these two problems. Thank you very much!

Solution

Unit place always 0 to 9.

Square of any number end with 0, 1, 4, 6, 9 this implies modolo 4 is 0, 1, -1.(Reminder 0, 1, -1)

similarly for modulo 8 is 0,1 or 4( reminder is 0, 1 or 4)