 # Units digit

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Units digit is the digit which denotes the 'number of ones' in that number. It is the rightmost digit in a number, preceding any decimal point.

A number abc denotes that it is a value equal to 100 x a + 10 x b + 1 x c. Therefore there are a hundreds, b tens and c ones. "Unity" means one and therefore, c is also called the units digit. For example in 3451, 1 is the units digit.

## Finding units digit

Generally a units digit can be identified by looking at the number and identifying the rightmost number before the decimal. But in some cases it is not so direct. In numbers with exponents the units digit has to be calculated. For example, to calculate the units digit of 2^60 it is advisable to use an indirect method rather than calculate the exact vale and then find the units digit.

Pattern or Cyclicity

When any number is raised to the power n, where n = 1, 2, 3..., its units digit follows a pattern or a cycle. For example, 2^1, 2^2, 2^3, 2^4... and so on end with 2, 4, 8, 6, 2, 4, 8, 6, 2, 4... In this case the units digit repeats after 4 powers. Therefore 2^1 will have the same units digit as 2^5, 2^9, 2^13... all 2^(4k+1), where k = 0, 1, 2, 3... The following table gives the patterns or cycles of all natural numbers from 1 to 9.

Number Cycle Pattern
1 1 1
2 4 2, 4, 8, 6
3 4 3, 9, 7, 1
4 1 1
5 1 5
6 1 6
7 4 7, 9, 3, 1
8 4 8, 4, 2, 6
9 2 9, 1

Remainder when divided by 10

Another method to find a units digit of a number is to find the numbers remainder when it is divided by 10. For example, 3456 when divided by 10 gives a remainder 6, which is the units digit. Remainders can be found using various methods.

### Relevant exercises

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