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Question

# 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 0≤r<5 ⇒ a2=(5m+r)2=25m2+r2+10mr [∵ (a+b)2=a2+2ab+b2] ⇒ a2=5(5m2+2mr)+r2 ...(i) Where, 0≤r<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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