Today I am going to share an interesting technique to solve Unit Digit questions.

Suppose you have a series

P, Q, R, S ,T, P, Q, R, S ,T,P, Q, R, S ,T,P, Q, R, S ,T,P, Q, R, S , T,P, Q, R, S, T .....

And you have to find out the 16

One way to solve this is by counting the 16

Why we have divided by 5 because the terms in the series are repeated after a cycle of 5.

Let us take another question.

Find out the 25

25/ 5 gives you the remainder zero (0)

In such case, your answer should be the last term of the cycle and the last term of the cycle is T

25

__ __ 8× _ _ 3 which means 8×3 = 2 4

Unit digit

So the unit digit in the product 268 × 453 is 4.

Remember in such questions, in such questions, you are only going to get concerned about the unit digits

3 in 753

3 in 43

6 in 1236

4 in 864

3 × 3 × 6 × 4 = ____ 6

So the unit digit in the product 753 × 43 × 1236 × 864 is 6.

From above table we can see that

In case unit digit is 2 or 3 or 7 or 8, it repeats itself after 4 cycles

Now let us pick up some questions based on this observation

###
Find the unit digit in 2

We know in case of 2, it repeats itself after a cycle of 4 . We will divide 49 by 4

49/4 remainder is 1

We write it as

2

That means the unit digit in the 2

###
Find the unit digit in 3

Solution: Now here the power is 52 and we know that in case of 3, it repeats itself after a cycle of 4 .

52/4 the remainder is 0.

In such cases, our answer should be the 4

So answer is unit digit in 3

Let us do some more complex examples

###
Find the last digit in the 7

Solution: In this case we need to divide the power by 4

The power is 453040

We know that a number is divisible by 4 if the number formed the last two digits is divisible by 4.

00/4 = 0 that means remainder is ZERO and we know that in case of 7 ,the cycle is 4 so we will find out the 4

7

7

= 9 × 9 (Unit digits)

=

From the table we also observe that

If the power is odd, the unit digit is 4 and if the power is even, the unit digit is 6

If the power is odd, the unit digit is 9 and if the power is even, the unit digit is 1

Let us do some of its applications

###

In 9

The answer is 4 × 1 = 4

Suppose you have a series

P, Q, R, S ,T, P, Q, R, S ,T,P, Q, R, S ,T,P, Q, R, S ,T,P, Q, R, S , T,P, Q, R, S, T .....

And you have to find out the 16

^{th}term of the series. How would you do this?One way to solve this is by counting the 16

^{th}term; you get your answer P.**The other way to solve**: You can divide the 16 by 5 and get the remainder as 1. So now answer would be the 1^{st}term that is P.Why we have divided by 5 because the terms in the series are repeated after a cycle of 5.

Let us take another question.

Find out the 25

^{th}term of the above series. Following the same procedure you get25/ 5 gives you the remainder zero (0)

In such case, your answer should be the last term of the cycle and the last term of the cycle is T

25

^{th}term is T.### Find out the unit digit in 268 × 453?

Now to solve this question, you are going to pick up the only last digits and in this case__ __ 8× _ _ 3 which means 8×3 = 2 4

Unit digit

So the unit digit in the product 268 × 453 is 4.

Remember in such questions, in such questions, you are only going to get concerned about the unit digits

### What is the unit digit in the product 753 × 43 × 1236 × 864?

Solution: let us pick up the unit digits and multiply them3 in 753

3 in 43

6 in 1236

4 in 864

3 × 3 × 6 × 4 = ____ 6

**(concern only about unit digit in the product)**So the unit digit in the product 753 × 43 × 1236 × 864 is 6.

**Now let us observe the pattern in the cycle of different digit in other words after how many cycles the last digit repeats itself.**2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

2^{1}=2 | 3^{1}=3 | 4=4^{1} | 5^{1}=5 | 6^{1}=6 | 7^{1}=7 | 8^{1}=8 | 9^{1}=9 |

2^{2}=4 | 3^{2}=9 | 4^{2}=6 | 5^{2}=5 | 6^{2}=6 | 7^{2}=9 | 8^{2}=4 | 9^{2}=1 |

2^{3}=8 | 3^{3}=7 | 4^{3}=4 | 5^{3}=5 | 6^{3}=6 | 7^{3}=3 | 8^{3}=2 | 9^{3}=9 |

2^{4}=6 | 3^{4}=1 | 4^{4}=6 | 5^{4}=5 | 6^{4}=6 | 7^{4}=1 | 8^{4}=6 | 9^{4}=1 |

2^{5}=2 | 3^{5}=3 | 4^{5}=4 | 5^{5}=5 | 6^{5}=6 | 7^{5}=7 | 8^{5}=8 | 9^{5}=9 |

2^{6}=4 | 3^{6}=9 | 4^{6}=6 | 5^{6}=5 | 6^{6}=6 | 7^{6}=9 | 8^{6}=4 | 9^{6}=1 |

2^{7}=8 | 3^{7}=7 | 4^{7}=4 | 5^{7}=5 | 6^{7}=6 | 7^{7}=3 | 8^{7}=8 | 9^{7}=9 |

From above table we can see that

In case unit digit is 2 or 3 or 7 or 8, it repeats itself after 4 cycles

Now let us pick up some questions based on this observation

###
Find the unit digit in 2^{49}?

We know in case of 2, it repeats itself after a cycle of 4 . We will divide 49 by 449/4 remainder is 1

We write it as

2

^{49}= 2^{1}= 2That means the unit digit in the 2

^{49 }is 2.###
Find the unit digit in 3^{52}.

Solution: Now here the power is 52 and we know that in case of 3, it repeats itself after a cycle of 4 .52/4 the remainder is 0.

In such cases, our answer should be the 4

^{th}powerSo answer is unit digit in 3

^{4}is 1.Let us do some more complex examples

###
Find the last digit in the 7^{45304000}

Solution: In this case we need to divide the power by 4The power is 453040

**00**We know that a number is divisible by 4 if the number formed the last two digits is divisible by 4.

00/4 = 0 that means remainder is ZERO and we know that in case of 7 ,the cycle is 4 so we will find out the 4

^{th}power of 77

^{4}if you still find difficult, let us simplify it7

^{4}= 7^{2 }× 7^{2}= 9 × 9 (Unit digits)

=

**8****1 Unit digit so the last digit in the 7**^{45304000}is 1From the table we also observe that

__in case of 4__If the power is odd, the unit digit is 4 and if the power is even, the unit digit is 6

__And same is the case with 9__If the power is odd, the unit digit is 9 and if the power is even, the unit digit is 1

Let us do some of its applications

###
**Find out the unit digit in 4**^{39 }×** 9**^{78} ? In 4^{39} the unit digit is 4 (the power is odd)

In 9^{39 }

^{78}?

^{78}the unit digit is 1 ( the power is even )The answer is 4 × 1 = 4