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Question

Show that the square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer q.

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Solution

It is known that any positive integer can be written in the form of 6m,6m+1,6m+2,6m+3,6m+4,6m+5 for some integer m.
Thus, an odd positive integer can be of the form 6m+1,6m+3,6m+5
We have, (6m+1)2=36m2+12m+1=6(6m2+2m)+1=6q+1,

where q=6(m2+2m) is an integer

Consider (6m+3)2=36m2+36m+9=36m2+36m+6+3=6(6m2+6m+1)+3=6(6m2+6m+1)+3=6q+3,
where q=6m2+6m+1 is an integer

Consider (6m+5)2=36m2+60m+25=36m2+60m+24+1=6(6m2+10m+4)+1=6q+1,
where q=6m2+10m+4 is an integer


Thus, the square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer q


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