Question

Prove that the square of any positive integer is of the form $5q$, $5q+1$, $5q+4$ for some integer $q$.

Answer 1

STEP 1 : Assumption

Let $x$ be any positive integer. When we divide x by 5, the remainder is either 0 or 1 or 2 or 3 or 4.

So, x can be written as x = 5m, or x = 5m +1 or x = 5m +2 or x = 5m +3 or x 5m + 4. Thus, we have the following cases:

CASE I When x = 5m: In this case, 25m² = 5 (5m²) = 5q, where q = 5m

CASE II When x = 5m + 1: In this case, x = (5m+ 1) = 25m + 10m +1 = 5 (5m + 2m) +1 5q + 1, where q = 5m* + 2m

CASE III When x = 5m +2: In this case, x = (5m + 2) = 25m + 20m + 4 = 5(5m2 +4m) + 4 = 5q +4, where q = 5m + 4m

CASE IV When x = 5m +3: In this case, x = (5m +3) = 25m2 + 30m +9 = (5m2 +30m + 5) +4 5 (5m +6m + 1) +4 = 5q+ 4, where q = 5m² + 6m + 1

CASE V When x = 5m + 4: In this case, = (5m + 4)2 25m2 + 40m + 16 5(5m+8m + 3) +1 5q + 1 where q = 5m + 8m + 3

Hence, x is of the form 5q or 5q + 1, 5q + 4. So, it cannot be of the form 5q +2 or 5q +3.a