Prove that the square of any positive integer is of the form 4q or 4q

1.	Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.
Answer» By Euclid’s division Algorithm a = bm + r, where 0 ≤ r ≤ b Put b = 4 a = 4m + r, where 0 ≤ r ≤ 4 If r = 0, then a = 4m If r = 1, then a = 4m + 1 If r = 2, then a = 4m + 2 If r = 3, then a = 4m + 3 Now, (4m)² = 16m² = 4 × 4m² = 4q where q is some integer (4m + 1)² = (4m)² + 2(4m)(1) + (1)² = 16m² + 8m + 1 = 4(4m² + 2m) + 1 = 4q + 1 where q is some integer (4m + 2)² = (4m)² + 2(4m)(2)+(2)² = 16m² + 24m + 9 = 16m² + 24m + 8 + 1 = 4(4m² + 6m + 2) + 1 = 4q + 1, where q is some integer Hence, the square of any positive integer is of the form 4q or 4q + 1 for some integer m

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

Answer»

By Euclid’s division Algorithm
a = bm + r, where 0 ≤ r ≤ b
Put b = 4
a = 4m + r, where 0 ≤ r ≤ 4
If r = 0, then a = 4m
If r = 1, then a = 4m + 1
If r = 2, then a = 4m + 2
If r = 3, then a = 4m + 3
Now, (4m)² = 16m²
= 4 × 4m²
= 4q where q is some integer
(4m + 1)² = (4m)² + 2(4m)(1) + (1)²
= 16m² + 8m + 1
= 4(4m² + 2m) + 1
= 4q + 1 where q is some integer
(4m + 2)² = (4m)² + 2(4m)(2)+(2)²
= 16m² + 24m + 9
= 16m² + 24m + 8 + 1
= 4(4m² + 6m + 2) + 1
= 4q + 1, where q is some integer

Hence, the square of any positive integer is of the form 4q or 4q + 1 for some integer m

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