show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q?
show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q?
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Let positive integer a = 4m + r ,
By division algorithm we know
here 0 ≤ r < 4 , So
When r = 0
a = 4m
Squaring both side , we get
a 2 = ( 4m ) 2
a 2 = 4 ( 4m 2 )
a 2 = 4 q , where q = 4m 2
When r = 1
a = 4m + 1
squaring both side , we get
a 2 = ( 4m + 1) 2
a 2 = 16m 2 + 1 + 8m
a 2 = 4 ( 4m 2 + 2m ) + 1
a 2 = 4q + 1 , where q = 4m 2 +
2m
When r = 2
a = 4m + 2
Squaring both hand side , we
get
a 2 = ( 4m + 3) 2
a 2 = 16m 2 + 9 + 24m
a 2 = 16m 2 + 24m + 8 + 1
a 2 = 4 ( 4m 2 + 6m + 2) + 1
a 2 = 4q + 1 , where q = 4m 2 +
6m + 2
Hence
Square of any positive integer
is in form of 4q or 4q + 1 ,
where q is any integer..
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