# The units digit of a triangular number iw ## 0, 1, 3, 5, 6 ##?

• Math100
In summary, the units digit of a triangular number can only be one of the following: 0, 1, 3, 5, 6, or 8. This is due to the fact that when a triangular number is divided by 10, the remainder can only be 0, 1, 3, 6, or 5, depending on the value of n. This can be seen by looking at the pattern of the units digit for different values of n. Therefore, the statement given in the homework is proven to be true.
Math100
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 ##.

Delta2
Math100 said:
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 ##.
Right. I think it is time to proceed to the next paragraph now, something more challenging.

Delta2 and Math100

## What is a triangular number?

A triangular number is a number that can be represented as a triangle of dots or objects. The first few triangular numbers are 1, 3, 6, 10, 15, and so on.

## Why is the units digit of a triangular number only 0, 1, 3, 5, or 6?

This is because the units digit of a triangular number is determined by the pattern of odd and even numbers in the sequence. Since triangular numbers are formed by adding consecutive numbers, the units digit will repeat in a cycle of 0, 1, 3, 5, and 6.

## How can I determine the units digit of a triangular number?

To find the units digit of a triangular number, you can use the formula (n(n+1))/2, where n is the number of the triangular number. Then, you can simply look at the units digit of the result to determine the units digit of the triangular number.

## Are there any exceptions to the pattern of the units digit of a triangular number?

Yes, there are a few exceptions to the pattern. For example, the triangular numbers 12, 22, and 32 have a units digit of 4 instead of 2. This is because these numbers are not formed by adding consecutive numbers, but rather by adding two consecutive triangular numbers.

## What is the significance of the units digit of a triangular number?

The units digit of a triangular number can be used to identify patterns and relationships between numbers. It can also be used in various mathematical calculations and problem-solving strategies.

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