Show that square of any positive odd integer is in form of 6q +1 and 6q+3 where q is some integer?
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Answer:we know that any positive integer can be of the firm 6m,6m+1,6m+2, 6m+3,, 6m+4 or 6m+5 for some integer m, So an odd positive integer x is of form 6m+1 or 6m+3
CASE 1- when x=6m+1 in this case
x²=(6m+1)²=36m²+12m+1=6(6m²+2m)+1
=6q+1 where q=6m²+m
CASE I I- when x=6m+3
x²=(6m+3)²=36m²+36m+9
=(36m²+36m+6)+3
6(6m²+6m+1)+3=6q+3,where q=6m²+6m+3
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