Question

use Euclid division lemma to show that the square of any positive integer cannot be of the form 5m+2or 5m+3 for some integer m.

Answer 1

similarly prove square of any positive integer cannot be of the form 5m+2or 5m+3 for some integer m.

Answer 2

Let n be any positive integer.

By Euclid’s division lemma,

n = 5q + r, 0 ≤ r < 5

n = 5q, 5q + 1, 5q + 2, 5q + 3 or 5q + 4

Hence

n^2 = (5q)^2 = 25q^2 = 5(5q^2) = 5m

n^2 = (5q + 1)^2 = 25q^2 + 10q + 1 = 5m + 1

n^2 = (5q + 2)^2 = 25q^2 + 20q + 4 = 5m + 4

Similarly,

n^2 = (5q + 3)^2 = 25q^2 + 30q + 5 + 4

= 5m + 4  and n2 = (5q + 4)2 = 5m + 1

Square of any positive integer cannot be  of the form 5m + 2 or 5m + 3