## Last two digits of a number ( Method of finding Last two digits)

I am dividing this method into four parts and we will discuss each part one by one:

a. Last two digits of numbers which end in one

b. Last two digits of numbers which end in 3, 7 and 9

c. Last two digits of numbers which end in 2

d. Last two digits of numbers which end in 4, 6 and 8

Before we start, let me mention binomial theorem in brief as we will need it for our calculations.

**Last two digits of numbers ending in 1**

Let’s start with an example.

**What are the last two digits of 31 ^{786}?**

Solution: 31** ^{786}** = (30 + 1)

**=**

^{786}**C**

^{786}** 1**

_{0 }×**+**

^{786}**C**

^{786}** 1**

_{1 }×

^{785}**×** (30) +

**C**

^{786}** 1**

_{2}×

^{784}**×** 30

**+ …, Note that all the terms after the second term will end in two or more zeroes. The first two terms are**

^{2}**C**

^{786}** 1**

_{0 }×**and**

^{786}**C**

^{786}** 1**

_{1 }×**× (30). Now, the second term will end with one zero and the tens digit of the second term will be the product of 786 and 3 i.e. 8. Therefore, the last two digits of the second term will be 80. The last digit of the first term is 1. So the last two digits of 31**

^{785}**are 81.**

^{786}Now, here is the shortcut:

**Find the last two digits of 41 ^{2789}**

In no time at all you can calculate the answer to be 61 (4 **×** 9 = 36). Therefore, 6 will be the tens digit and one will be the units digit)

**Find the last two digits of 71 ^{56747}**

Last two digits will be 91 (7 **×** 7 gives 9 and 1 as units digit)

Now try to get the answer to this question within 10 s:

**Find the last two digits of 51 ^{456} **

**×**

**61**

^{567}The last two digits of 51** ^{456}** will be 01 and the last two digits of 61

**will be 21. Therefore, the last two digits of 51**

^{567}

^{456}**×** 61

**will be the last two digits of 01**

^{567}**×**21 = 21

**Last two digits of numbers ending in 3, 7 or 9**

**Find the last two digits of 19 ^{266}.**

19** ^{266}** = (19

**)**

^{2}**. Now, 19**

^{133}**ends in 61 (19**

^{2}**= 361) therefore, we need to find the last two digits of (61)**

^{2}**.**

^{133}Once the number is ending in 1 we can straight away get the last two digits with the help of the previous method. The last two digits are 81 (6 **×**3 = 18, so the tens digit will be 8 and last digit will be 1)

**Find the last two digits of 33 ^{288}.**

33** ^{288}** = (33

**)**

^{4}**. Now 33**

^{72}**ends in 21 (33**

^{4}**= 33**

^{4}

^{2}**×**33

**= 1089**

^{2}**×**1089 = xxxxx21) therefore, we need to find the last two digits of 21

**. By the previous method, the last two digits of 21**

^{72}**= 41 (tens digit = 2**

^{72}**×**2 = 4, unit digit = 1)

So here’s the rule for finding the last two digits of numbers ending in 3, 7 and 9:

Now try the method with a number ending in 7:

**Find the last two digits of 87 ^{474}.**

87** ^{474}** = 87

^{472}**×**87

**= (87**

^{2}**)**

^{4}

^{118}**×**87

**= (69**

^{2}**×**69)

^{118}**×**69 (The last two digits of 87

**are 69) = 61**

^{2}

^{118}**×** 69 = 81

**×** 69 = 89

If you understood the method then try your hands on these questions:

**Find the last two digits of:**

- 27
^{456}

2. 79^{83}

3. 583^{512}

**Last two digits of numbers ending in 2, 4, 6 or 8**

There is only one even two-digit number which always ends in itself (last two digits) – 76 i.e. 76 raised to any power gives the last two digits as 76. Therefore, our purpose is to get 76 as last two digits for even numbers. We know that 24** ^{2}** ends in 76 and 2

**ends in 24. Also, 24 raised to an even power**

^{10}**always ends with 76 and 24 raised to an odd power always ends with 24. Therefore, 24**

**will end in 76 and 24**

^{34}**will end in 24.**

^{53}Let’s apply this funda:

**Find the last two digits of 2 ^{543}.**

2** ^{543}** = (2

**)**

^{10}

^{54}**×**2

**= (24)**

^{3}**(24 raised to an even power)**

^{54}**×**2

**= 76**

^{3}**×**8 = 08

(**NOTE: **Here if you need to multiply 76 with 2** ^{n}**, then you can straightaway write the last two digits of 2

**because when 76 is multiplied with 2**

^{n}**the last two digits remain the same as the last two digits of 2**

^{n}**. Therefore, the last two digits of 76**

^{n}**×**2

**will be the last two digits of 2**

^{7}**= 28)**

^{7}Same method we can use for any number which is of the form 2** ^{n}**. Here is an example:

**Find the last two digits of 64 ^{236}.**

64** ^{236}** = (2

**)**

^{6}**= 2**

^{236}**= (2**

^{1416}**)**

^{10}

^{141}**×**2

**= 24**

^{6}**(24 raised to odd power)**

^{141}**×**64 = 24

**×**64 = 36

Now those numbers which are not in the form of 2n can be broken down into the form 2n ´ odd number. We can find the last two digits of both the parts separately.

Here are some examples:

**Find the last two digits of 62 ^{586}.**

62** ^{586}** = (2

**×**31)

**= 2**

^{586}

^{586}**×**3

**= (2**

^{586}**)**

^{10}

^{58}**×** 2

^{6}**×**31

**= 76**

^{586}**×**64

**×**81 = 84

**Find the last two digits of 54 ^{380}.**

54** ^{380}** = (2

**×**3

**)**

^{3}**= 2**

^{380}

^{380}**×**3

**= (2**

^{1140}**)**

^{10}

^{38}**×**(3

**)**

^{4}**= 76**

^{285}**×**81

**= 76**

^{285}**×**01 = 76.

**Find the last two digits of 56 ^{283}.**

56** ^{283}** = (2

^{3}**×**7)

**= 2**

^{283}

^{849}**×**7

**= (2**

^{283}**)**

^{10}

^{84}**×**2

^{9}**×**(7

**)**

^{4}

^{70}**×**7

**= 76**

^{3}**×**12

**×**(01)

^{70}**×**43 = 16

**Find the last two digits of 78 ^{379}.**

78** ^{379}** = (2

**×**39)

**= 2**

^{379}

^{379}**×**39

**= (2**

^{379}**)**

^{10}

^{37}**×**2

^{9}**×**(39

**)**

^{2}

^{189}**×**39 = 24

**×**12

**×**81

**×**39 = 92

Now try to find the last two digits of

1. 34^{576}

2. 28^{287}

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Thnks for these types of Question. Really helps a lot.

It is very helpfull for competitive exams.

Hi sir….you explained this concepts very well.

But one concept is missing.

How to find last two digits of nunber ending with 5. A number ending with 5(odd) cannot be converted into another number ending with 1.

Please also add the concept of this.

For numbers ending with 5, its pretty straight forward

If your power is even, then your last 2 digits are always 25

And if the power is odd, then look at your 10s digit.

If the 10s digit is odd, then last 2 digits are 75

If the 10s digit is even, then last 2 digits are 25

That’s about it

THIS ARTICLE WAS REALLY VERY GUD.THANKYOU SIR