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Question

Show that the square of any odd integer is of the form 4q+1, for some integer q.


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Solution

Proving that the square of any odd integer is of the form 4q+1, for some integer q.

Apply Euclid's division algorithm, a=bm+r where a,b,m,r are non-negative integers and 0r<b.

Assume that b=4.

Substitute r=0,1,2,3 and b=4 into a=bm+r.

a=4mr=0a=4m+1r=1a=4m+2r=2a=4m+3r=3

We know that a is odd

So, the value of a cannot be 4mand4m+2 since 4mand4m+2 are divisible by 2 and so are even.

Thus, a=4m+1anda=4m+3.

Square a=4m+1 on both sides.

a2=4m+12a2=16m2+8m+1a2=44m2+2m+1a2=4q+1qissomeintegerandq=4m2+2m

Square a=4m+3 on both sides.

a2=4m+32a2=16m2+24m+9a2=44m2+6m+2+1a2=4q+1qissomeintegerandq=4m2+6m+2

Hence, it is proved that the square of any odd integer is of the form 4q+1, for some integer q.


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