# CAT Exam Notes: How to find the last two digits

Finding out last two digits of a number is a skill that will help you not only solve some tough questions but quickly eliminate a few wrong options as well. Also, it is a skill that you can develop and master in less than 15 minutes. All you need to do is to remember a few rules and the rest will follow.

**Funda 1: A number ending in 1**

*Last two digits of (…a1)^(…b) will be [Last digit of a*b]1*

Ex 1.1 Last two digits of 491^83 = [Last digit of 9*3]1 = [Last digit of 27]1 = 71

Ex 1.2 Last two digits of 571^64 = [Last digit of 7*4]1 = [Last digit of 28]1 = 81

** ****Funda 2: Solving last two digits for odd numbers**

*Change the odd number to something that ends in 1.*

Ex 2.1 What are the last two digits of (86789)^41?

For finding out the last two digits of an odd number raised to a power, we should first try and reduce the base to a number ending in 1.

After that, we can use the property, last two digits of (…a1)^(…b) will be [Last digit of a*b]1

Let us try and apply this concept in the given question

Last two digits of (86789)^41

= Last two digits of 89^41

= Last two digits of 89 * 89^40

= Last two digits of 89 * (..21)^20

= Last two digits of 89 * 01 {Here I have used the concept mentioned above} = 89

**Funda 3: Solving last two digits for even numbers**

*24 raised to an odd power will end in 24 and even power will end in 76*

This comes in handy when you are calculating last two digits of an even number or a power of 2. Do keep in mind that 2^10 is 1024

Let me take an example to explain this concept further

Ex 3.1 Find last two digits of 1456^72

1456^72 = 16^72 x 91^72

Last two digits of 16^72 = Last two digits of 2^208 = Last two digits of 2^20 x 2^8 = Last two digits of 1024^2 x 256 = Last two digits of 76 x 56 = 56

Last two digits of 91^72 = [Last digit of 9*2]1 = [Last digit of 18]1 = 81

Overall the last two digits = Last two digits of 56 x 81 = 36

The point to note here is that I converted the given even number into powers of two and an odd number and then solved it.

**Funda 4: Last two digits of a number ending in 5**

*If the second last digit of the base and the power, both are odd – it will end in 75; otherwise the last two digits will be 25.** *

Ex 4.1 Last two digits of 75^65 = 75 (Second last digit of the base and the power – both odd)

Ex 4.2 Last two digits of 65^75 = 25

Ex 4.3 Last two digits of 35^73 = 75 (Second last digit of the base and the power – both odd)

Ex 4.4 Last two digits of 1995^2014 = 25

Ex 4.5 Last two digits of 1995^2015 = 75 (Second last digit of the base and the power – both odd)

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