Prove that the square of any odd integer is of the form 8n+1

Question

Reagan Huffman

Mathematics

28 February, 11:05

Prove that the square of any odd integer is of the form 8n+1

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Sarahi Lucas · Answer 1

For any integer x,

if x is even, then

x^2+x = (2k) ^2 + 2k = 2 (2k^2 + k) is even

if x is odd, then

x^2+x = (2k+1) ^2 + 2k+1 = 2 (2k^2 + 3k + 1) is also even.

When x is odd,

x^2 = (2k+1) ^2 = 4 (k^2+k) + 1 = 4 (2n) + 1 = 8n+1