Let <TEX>${\sigma}_s(N)$</TEX> denote the sum of the s-th power of the positive divisors of N and <TEX>${\sigma}_{s,r}(N;m)={\sum_{d{\mid}N\\d{\equiv}r\;mod\;m}}\;d^s$</TEX> with <TEX>$N,m,r,s,d{\in}\mathbb{Z}$</TEX>, <TEX>$d,s$</TEX> > 0 and <TEX>$r{\geq}0$</TEX>. In a celebrated paper [33], Ramanuja proved <TEX>$\sum_{k=1}^{N-1}{\sigma}_1(k){\sigma}_1(N-k)=\frac{5}{12}{\sigma}_3(N)+\frac{1}{12}{\sigma}_1(N)-\frac{6}{12}N{\sigma}_1(N)$</TEX> using elementary arguments. The coefficients' relation in this identity (<TEX>$\frac{5}{12}+\frac{1}{12}-\frac{6}{12}=0$</TEX>) motivated us to write this article. In this article, we found the convolution sums <TEX>$\sum_{k</TEX><TEX>&</TEX><TEX>lt;N/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(N-mk;2)$</TEX> for odd and even divisor functions with <TEX>$i,j=0,1$</TEX>, <TEX>$m=1,2,4$</TEX>, and <TEX>$d{\mid}m$</TEX>. If N is an odd positive integer, <TEX>$i,j=0,1$</TEX>, <TEX>$m=1,2,4$</TEX>, <TEX>$s=0,1,2$</TEX>, and <TEX>$d{\mid}m{\mid}2^s$</TEX>, then there exist <TEX>$u,a,b,c{\in}\mathbb{Z}$</TEX> satisfying <TEX>$\sum_{k&</TEX> <TEX>lt;2^sN/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(2^sN-mk;2)=\frac{1}{u}[a{\sigma}_3(N)+bN{\sigma}_1(N)+c{\sigma}_1(N)]$</TEX> with <TEX>$a+b+c=0$</TEX> and (<TEX>$u,a,b,c$</TEX>) = 1(Theorem 1.1). We also give an elementary problem (O) and solve special cases of them in (O) (Corollary 3.27).