Question

find two positive odd number,sum of whose square is 74?

Answer 1

The no.s are-
7 and 5
Since 7²+5²=49+25=74.
By trial and error method.

Answer 2

Let the required numbers be $2m+1$ and $2n+1$
$(2m+1)^2+(2n+1)^2=74$
$4m^2+4m+1+4n^2+4n+1=74$
$m^2+m+n^2+n=18$

Clearly $0\le\,m,n\,\le\,3$

By working the four cases and by symmetry $m,n$ are $(m,n)=(2,3),(3,2)$
therefore the required numbers are $2*2+1, 2*3+1 = 5,7$