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Question 3
Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

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Solution

Let a be an arbitrary positive integer.
Then, by Euclid's division algorithm, corresponding to the positive integers a and 5, there exist non - negative integers m and r such that:
a=5m+r, where 0r<5
a2=(5m+r)2=25m2+r2+10mr [ (a+b)2=a2+2ab+b2]
a2=5(5m2+2mr)+r2 ...(i)
Where, 0r<5

Case 1
When r = 0, putting r = 0 in eq. (i) we get,
a2=5(5m2)=5q
where, q=5m2 is an integer.

case II
When r = 1, putting r = 1 is eq. (i) we get,
a2=5(5m2+2m)+1
a2=5q+1
Where, q=(5m2+2m) is an integer.

Case III
When r = 2, putting r = in eq. (i) we get,
a2=5(5m2+4m)+4=5q+4
Where, q=(5m2+4m)+4=5q+4

Case IV
When r = 3, putting r = 3 in eq (i) we get,
a2=5(5m2+6m)+9=5(5m2+6m)+5+4
=5(5m2+6m+1)+4=5q+4
Where, q = (5m2+6m+1) is an integer.

Case V
When r = 4, putting r = 4 in eq (i), we get,
a2=5(5m2+8m)+16=5(5m2+8m)+15+1
a2=5(5m2+8m+3)+1=5q+1
Where, q=(5m2+8m+3) is an integer.
Hence, the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.


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