using euclids division lemma show that the square of any positive integer is of the 5m, 5m+1 or 5m+4 for some integer m
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using euclids division lemma show that the square of any positive integer is of the 5m, 5m+1 or 5m+4 for some integer m
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Solution:-
Let 'a' be any positive integer and b = 5.
Applying Euclid's division lemma with a and b,we get,
a =5q+r,where 0<=r<5 and q is some integer.
Therefore, r =0,1,2,3,4
When ,r =0,1,2,3,4,then,
a =5q,5q+1,5q+2,5q+3,5q+4.
For some integer m ,square of positive integers is given by:
Now,When a =5q
=>a^2 = (5q)^2
= 25q^2
= 5×5q^2
= 5m , where m=5q^2.
When, a = 5q+1
=>a^2 = (5q+1)^2
=25q^2+10q+1
=5(5q^2+2q)+1
=5m+1, where m =(5q^2+2q)
Similarly, Do upto a =5q +4
Therefore, Square of any positive integer is of the form 5m,5m+1 or 5m+4,for some integer m.
Hope it helps you.
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