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

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

Answers (1)

By Euclid’s division –
Any positive integer can be written as:
A = bm + r
Here b = 5
r is remainder when we divide A by 5, therefore:
0 \leq r < 5, r = 0, 1, 2, 3.4
A = 5m + r                  …(1)

Case 1:
A = 5m  
(5m)2 = 25 m2
= 5(5m2)
= 5q    
(Here q = 5 m2)

Case 2:
A = 5m + 1
  (5m + 1)2 = 25 m2 + 1 + 10 m
25m2 + 10 m + 1
5(5m2 + 2m) + 1
5q + 1 
(Here q = 5m2 + 2m)
Similarly, we can verify it for 5m + 2, 5m + 3, 5m + 4
Here, square of any positive integer is in the form of 5q, 5q + 1
Hence, square of any positive integer cannot be of the form 5q + 2 or 5q + 3.

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