A number theory problem by Nutan Sinha

We need to find $\large \color{#D61F06}22^3 + \color{#3D99F6} 23^3 + \cdots + \color{#20A900}55^3 + \cdots + \color{#3D99F6} 87^3 + \color{#D61F06} 88^3 \color{#333333} \pmod{110}$

Note that $a+b$ divides $a^n+b^n$ for every odd $n$ , therefore combining extreme terms, they will be congruent to $0 \pmod{110}$ , however , the middle-most term which is $55^3$ , will be left uncombined , and $55^3 \equiv 55 \pmod{110}$ , therefore whole sum will leave a remainder of $\boxed{55}$ when divided by $110$