Question

.......... this question is totally going over my head.. -_-|| so if u guys are able to understand the question below please help me.... -_-# ..que]-prove that square of any positive integer is in the form of 4q or 4q+1.(￣へ￣)

thanks...
if you understand this please help me..
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Answer 1

Step-by-step explanation:

Applying Euclid's division algorithm with a and b = 4.

a = 4Q + r, 0  r < 4

When r = 0, a = 4Q  a2 = 16Q2 = 4(4Q2) = 4q, where q = 4Q2

 When r = 1, a = 4Q + 1

 a2 = (4Q + 1)2 = 16Q2 + 1 + 8Q = 4Q(4Q + 2) + 1 = 4q + 1, where q = Q(4Q + 2)

When r = 2, a = 4Q + 2

 a2 = (4Q + 2)2 = 16Q2 + 4 + 16Q = 4(4Q2 + 4Q + 1) = 4q,

(where q = 4Q2 + 4Q + 1 )

When r = 3, a = 4Q + 3

 a2 = (4Q + 3)2 = 16Q2 + 9 + 24Q = 4(4Q2 + 6Q + 2) + 1 = 4q + 1,

(where q = 4Q2 + 6Q + 2 )

Hence, the square of any positive integer is of the form 4q or 4q + 1 for some integer q.