Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Question

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Monjur Alahi

5 views

2025-01-04 04:42

2 Answers

Salim Ahmed Kawsar

Let N = 5p + 1.

Then,

According to the condition:

N²= 25p² + 10p + 1 ⇒ 5(5p² + 2p) + 1 ⇒ 5A+1

Where A = 5p² + 2p

Therefore N² is of the form 5m + 1.

5 views Answered 2023-02-05 15:29

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

Let n = 5q + 1 where q is a positive integer
∴ n² = (5q + 1)²
= 25q² + 10q + 1
= 5(5q² + 2q) + 1
= 5m + 1, where m is some integer
Hence, the square of any positive integer of the form 5q + 1 is of the same form.

5 views Answered 2023-02-05 15:29