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Show that any square of an odd positive integer is of the form 5q ,5q+1,5q+4 for some integer q

Posted by Husita Patel 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

Any integer can be written in the form 5m , 5m+1, 5m+2 (where m is any integer)

(5m)² = 25m² = 5 × 5m² = 5q (where q = 5m²)

(5m+1)² = (5m)² + 2 ×5m × 1 + 1² [by using identity- (a+b)²= (a²+ 2ab +b² )]
   = 25m² + 10 m + 1
= 5 (5m²+ 2m) +1
= 5q +1 ( where q= 5m²+2m)
(5m+2)²= (5m)² + 2× 5m × 2 +2² [by using identity- (a+b)²= (a²+ 2ab +b² )]
   = 25m² + 20 m +4
   = 5 (5m²+4m ) + 4
= 5q+4 (where q= 5m²+4m)
Hence proved

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