Unit Digit Shortcut - Find Last digit of any number

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^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 get

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^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 them

3 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
21=2	31=3	41=4	51=5	61=6	71=9	81=4	91=9
22=4	32=9	42=6	52=5	62=6	72=3	82=2	92=1
23=8	33=7	43=4	53=5	63=6	73=1	83=6	93=9
24=6	34=1	44=6	54=5	64=6	74=7	84=8	94=1
25=2	35=3	45=4	55=5	65=6	75=9	85=4	95=9
26=4	36=9	46=6	56=5	66=6	76=3	86=8	96=1
27=8	37=7	47=4	57=5	67=6	77=1	87=4	97=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 2⁴⁹?

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

2⁴⁹= 2¹= 2

That means the unit digit in the 2⁴⁹ is 2.

Find the unit digit in 3⁵².

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 power

So answer is unit digit in 3⁴is 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 4

The power is 45304000

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 7

7⁴if you still find difficult, let us simplify it

7⁴= 7² × 7²

= 9 × 9 (Unit digits)

= 81 Unit digit so the last digit in the 7^45304000 is 1

From 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 4³⁹ × 9⁷⁸ ? In 4³⁹ the unit digit is 4 (the power is odd)

In 9⁷⁸ the unit digit is 1 ( the power is even )

The answer is 4 × 1 = 4

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