Question

Prove that square of any positive integer is of the form 5m,5m+1,5m+1 for some integer n

Answer 1

Prove that square of any positive integer is of the form 5m,5m+1,5m+1 for some integer m

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Solution:-

Let the three positive integers be 5q, 5q+1 and 5q+2.

Now, by squaring of these integers...

(5q)²

=> 25q²

=> 5(5q²)

=> 5m

where, m = 5q²

(5q+1)²

=> (5q)² + (1)² + 2×5q×1

=> 25q² + 1 + 10q

=> 25q² + 10q + 1

=> 5(5q² + 2q) + 1

=> 5m + 1

where, m = 5q² + 2q

(5q+2)²

=> (5q)² + (2)² + 2×5q×2

=> 25q² + 4 + 20q

=> 25q² + 20q + 4

=> 5(5q² + 4q) + 4

=> 5m + 4

where, m = 5q² + 4q

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Hence proved, square of any positive integer is of the form 5m, 5m+1 and 5m+4.

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