- #1

Math100

- 773

- 219

- Homework Statement
- Prove the following statement:

The units digit of a triangular number is ## 0, 1, 3, 5, 6 ##, or ## 8 ##.

- Relevant Equations
- None.

Proof:

Let ## t_{n} ## denote the ## nth ## triangular number such that ## t_{n}=\frac{n^{2}+n}{2} ## for ## n\geq 1 ##.

Then ## n\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ##, or ## 9\pmod {10} ##.

Thus ## t_{n}\equiv 0, 1, 3, 6, 10, 15, 21, 28, 36 ##, or ## 45\pmod {10} ##.

Therefore, the units digit of a triangular number is ## 0, 1, 3, 5, 6 ##, or ## 8 ##.

Let ## t_{n} ## denote the ## nth ## triangular number such that ## t_{n}=\frac{n^{2}+n}{2} ## for ## n\geq 1 ##.

Then ## n\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ##, or ## 9\pmod {10} ##.

Thus ## t_{n}\equiv 0, 1, 3, 6, 10, 15, 21, 28, 36 ##, or ## 45\pmod {10} ##.

Therefore, the units digit of a triangular number is ## 0, 1, 3, 5, 6 ##, or ## 8 ##.