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  • Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m

Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m

Answers (1)

By Euclid’s division –
Any positive integer can be written as:
A = bq + r
Here b = 6
r is remainder when we divide A by 5, therefore:
0 \leq r < 6,    r = 0, 1, 2, 3,4,5
A = 6q + r                   …(1)

Case 1:
A = 6q       
A2 = (6q)2 = 36q2 = 6(6q2)
A2 = 6m          
(m = 6q2)        

Case 2:
A = 6q + 1
 A2 = (6q + 1)2
A2 = 36q2 + 1 + 2(6q)
(Using (a + b)2 = a2 + b2 + 2ab)
A2 = 36q2 + 1 + 12q
A2 = 6(6q2 + 2q) + 1
A2 = 6m + 1    
(m = 6q2 + 2q)
Similarly, we can verify it for 6q + 2, 6q + 3, 6q + 4, 6q + 5
Now, it is clear that the square of any positive integer cannot be of the form      6m + 2, 6m + 5.    

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