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Use Euclid's division lemma to show …

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Use Euclid's division lemma to show that the square of any positive integer cannot be of the form 5m+2 or 5m+3 for some integer m.

Posted by Manvi Gupta 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

Let n be any positive integer.
By Euclid's division lemma, n = 5q + r, 0{tex}\leqslant{/tex}r < 5
n = 5q,5q + 1,5q + 2 ,5q 4- 3 or 5q + 4, where q{tex}\in \omega{/tex}
Now we find the square of n
If n=5q then (5q)² = 25q²= 5(5q²) = 5m
If n=5q+1then n2= (5q + 1 )² = 25q² + 10q + 1 = 5m + 1
If n=5q+2 then n² = (5q + 2)² = 25q² + 20q + 4 = 5m + 4
If n=5q+4 then n²=(5q + 3)²=25q²+30q+9=5m + 1
Thus square of any positive integer is in the form of 5m,5m+1 or 5m+4, hence cannot be of the form 5m + 2 or 5m + 3.

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