Question

Prove that square of any positive integer is of the form 5q,5q+1,5q+4 where q is some integer

Answer 1

Answer:

Pls see the photo and just change the term 'q' to 'p'

Answer 2

Answer:

SOLUTION :  

Since positive integer n is of the form of 5m or 5m + 1, 5m + 4.

Case : 1

If n = 5m , then

n² = (5m)²

[On squaring both sides]

n² = 25m²

n² = 5 (5m)

n² = 5q (Where q = 5m)

Case : 2

If n = 5m + 1, then

n² = (5m +1)²

[On squaring both sides]

n² = (5m)²+ 10m + 1

[(a+b)² = a² + b² + 2ab]

n² = 25m² + 10m + 1

n² = 5m (5m + 2) + 1

n² = 5q +1 , where q = m (5m + 2)

Case : 3

If n = 5m + 2, then

n² = (5m + 2)²

[On squaring both sides]

n² = (5m)² + 20m + 4

[(a+b)² = a² + b² + 2ab]

n² = 25m² + 20m + 4

n² = 5m (5m + 4) + 4

n² = 5q + 4 (where q = m (5m + 4))

Case : 4

If n = 5m + 4, then

n²= (5m + 4)²

[On squaring both sides]

n²= (5m)² + 40m + 4²

[(a+b)² = a² + b² + 2ab]

n² = 25m² + 40m + 16

n² = 5 (5m² + 8m + 3) + 1

n² = 5q + 1 , where q = 5m² + 8m + 3 )

Hence ,it is proved that the square of any positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q.

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