Triangular number modulo 3

Statement

Find the remainder of $T_{100}=\dfrac{100\cdot 101}{2}$ upon division by $3$.

Solution

We have $T_{100}=\dfrac{100\cdot 101}{2}=5050$. Since $5+0+5+0=10$ and $1+0=1$, the remainder mod $3$ is $T_{100}\equiv \boxed{1}\pmod 3.$ (General rule: $T_n\equiv 1\pmod 3$ iff $n\equiv 1\pmod 3$; here $100\equiv 1\pmod 3$.)