Question

Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

Answer 1

By Euclid’s division algorithm
a = bq + r, where 0 ≤ r ≤ b
Put b = 4
a = 4q + r, where 0 ≤ r ≤ 4
If r = 0, then a = 4q even
If r = 1, then a = 4q + 1 odd
If r = 2, then a = 4q + 2 even
If r = 3, then a = 4q + 3 odd
Now, (4𝑞 + 1)² = (4𝑞)² + 2(4𝑞)(1) + (1)²
= 16𝑞² + 8𝑞 + 1
= 8(2𝑞² + 𝑞) + 1
= 8m + 1 where m is some integer
Hence the square of an odd integer is of the form 8q + 1, for some integer q