show that the SQUARE of any positive integer is of the form 5q, 5q + 1 or 5q + 3 for some integer q
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I'm asking the same question more than 2-3 times
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Sol : Let x be any positive integer Then x = 5q or x = 5q+1 or x = 5q+4 for integer x.
If x = 5q, x2 = (5q)2 = 25q2 = 5(5q2) = 5n (where n = 5q2 )
If x = 5q+1, x2 = (5q+1)2 = 25q2+10q+1 = 5(5q2+2q)+1 = 5n+1
(where n = 5q2+2q )
If x = 5q+4, x2 = (5q+4)2 = 25q2+40q+16 = 5(5q2 + 8q + 3)+ 1 = 5n+1
(where n = 5q2+8q+3 )
∴in each of three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.
If x = 5q, x2 = (5q)2 = 25q2 = 5(5q2) = 5n (where n = 5q2 )
If x = 5q+1, x2 = (5q+1)2 = 25q2+10q+1 = 5(5q2+2q)+1 = 5n+1
(where n = 5q2+2q )
If x = 5q+4, x2 = (5q+4)2 = 25q2+40q+16 = 5(5q2 + 8q + 3)+ 1 = 5n+1
(where n = 5q2+8q+3 )
∴in each of three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.
rakhithakur:
and I answer your another question
x^2 : 5(5q^2+6q+?)+3
thanks for that alot
in which class do you study
why?
bcz as my opinion say it is question form class 10
is it right
the answer which I'm saying be like : add or subtract it come as 9 and subtracting 3 from that & it should be divisible by5
yes
and this one?...
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please see the attachment it is similar to your q
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