For finding the unit digit of expression, we have to understand the nature of expression. After that, we will decide how to find the unit digit of expression.

A given expression can be of the following two types:

- When the number is given in the form of the product of numbers.
- When the number is given in the form of index.

## Unit Digit Of Expression

Let’s see each of them in detail

### When the number is given in the form of the product of numbers

To find the unit’s digit in the product of two or more than two numbers, we take the unit’s digit of every number and then multiply them.

Then the unit’s digit of the resultant product is the unit’s digit of the product of the original numbers.

Example:

2208 X 35581 X 45273952 X 8532

here are four numbers and we have to find the unit digit of these numbers product.

Now we will multiply the unit digit of these numbers

= 8 X 1 X 2 X 2

= 32

Here the unit digit is 2.

So the unit digit of the product of the numbers (2208 X 35581 X 45273952 X 8532) will be 2.

### When the number is given in the form of index

Now we will find the unit digit when the number is given in the form of an index or expression.

Suppose a number is given in the form of A^{n}.

Then following cases arise:

#### Case 1. If n is a multiple of 4

(i) If A is an even number, i.e. 2, 4, 6, & 8, then the unit digit is 6.

(ii) If A is an odd number, i.e. 1,3,7,9, the unit digit is 1.

Examples:

(i) (4137)^{36} In this number unit digit is 1 because 4137 is divisible by 4 and this is an odd number.

(ii) (2138)^{756} In this number unit digit is 6 because 2138 is divisible by 4 and this is an even number.

#### Case 2. If n is not a multiple of 4

Let r be the remainder when n is divided by 4.

i.e. n = 4q+r

then the unit place of the A^{n} is equal to the unit place of the digit A^{r}.

For example, consider 7^{105}. Here 105 is not divisible by 4, so when 105 is divided by 4, we get the remainder as 1.

That’s by the unit digit in 7^{105} = the unit digit in 7^{1} = 7.

#### Case 3. If unit digit of A is 0, 1, 5 or 6.

If the unit digit of A is 0, 1, 5, or 6 then the resultant unit digit of An remains the same.

Examples:

- The unit digit in 8567
^{1041}is 7. - The unit digit in 7756
^{125}is 6. - The unit digit in 500
^{51}is 0. - The unit digit in 155
^{120}is 5.

#### Case 4. If the unit digit of A is 9 and Powe of A is even

If the unit digit of A is 9 and Powe of A is even, then the unit’s digit will be 1 and if the power of A is odd, then the unit digit will be 9.

Examples:

- (539)
^{140}Since the unit digit of A is 9 and power is even.

So, the unit digit is 1. - (539)
^{141}Since the unit digit of A is 9 and power is odd.

So, the unit digit is 9.

## Some Important Facts

Here are some important facts of unit digit of expressions

- Square of every odd number is always odd number while the square of every even number is always even.
- A number obtained be squaring a number does not have 2, 3, 7, 8 at its unit place.
- Sum of first n natural numbers = n(n+1)/2.
- Sum of first n odd numbers = n
^{2}. - Sum of first n even numbers n(n+1).
- Sum of squares of first n natural numbers = {n(n+1)(2n+1)}/6.
- Sum of cubes of first n natural numbers = [{n(n+1)}/2]
^{2}. - There are 15 prime numbers between 1 and 50, and 10 prime numbers between 50 and 100.
- If p divides q and r, then p divides their sum and difference also.
- Fon any natural number n, (n
^{3}-n) is divisible by 6. - (x
^{m}– a^{m}) is divisible by (x-a) for all values of m. - (x
^{m}– a^{m}) is divisible by (x+a) for all even values of m. - (x
^{m}+ a^{m}) is divisible by (x+a) for all odd values of m. - Number of prime factors of a
^{p}b^{q}c^{r}d^{s}is p+q+r+s, where a,b,c, and d are prime numbers.

Examples:

##### Q. **What is the Unit Digit of 6**^{15} -7^{4} – 9^{3}?

^{15}-7

^{4}– 9

^{3}?

Sol. The unit digit is **6 ^{15}** =

**6**

^{4X3+3}=

**6**= 6.

^{3}The unit digit is

**7**=

^{4}**7**= 1.

^{4×1}The unit digit is

**9**= 9.

^{3}Hence **the Unit Digit of 6 ^{15} -7^{4} – 9^{3}** = 6-1-9 = 5-9 = 15 – 9 = 6.

Here we can see that 9 can’t subtract from 5. So we will take carry over from the ten’s place.

**the Unit Digit of 6 ^{15} -7^{4} – 9^{3}** is 6.

Q. What is the unit digit of 7^{139}?

Sol. Here we can see that in 7^{139} number 139 is not divisible by 4.

Hence 7^{139} = 7^{3×34+3}

7^{139} = 7^{3}

7^{3} = 3

Hence the unit digit is 3.

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