# How to find last two digit of a large number in the form of power

You might have solved last year CAT papers(when it was available) or any AIMCAT/SIMCAT/..CAT.

**Quotlly**

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A very unique question that troubles you is the last 2 digit of a number **in form of a product** & last 2 digit of number **in form of power.**

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Here we are going to discuss the type of question where we need to find **last 2 digit of a number in form of power.**

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The most important thing in this type of question is that the number is so big that you think it can be only be solved by some magic & you actually don’t try such questions.

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**For e.g. Find last 2 digits of number 66^32 ?**

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Looks really tough. But don’t worry we will provide you step by step approach to solve this problem in a very easy manner.

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Step 1 : Let the number be a^b. Based on the unit’s digit of of a we will have 4 cases.

Case 1: The unit’s digit of a is 1.

In this case Unit’s digit = 1.

Ten’s digit = Unit’s digit of (Multiplication of ten’s digit of a with unit’s digit of b.)

For e.g. **Find last 2 digits of number 61^32 ?**

Unit’s digit = 1

Ten’s digit = Unit’s digit of 6 x 2 = 2.

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Case 2:The unit’s digit of a is 3, 7 or 9.

In this case we will convert base to such a number which ends in 1. After getting this number we will use case 1 to solve the problem.

For e.g. **Find last 2 digits of number 69^34 ?**

69^34 -> (69^2)^17 -> (..61)^17

So last 2 digit of 69^34 = 21.

**Find last 2 digits of number 69^33 ?**

69^33 = 69^32 x 69 = (..61)^16 x 69 = ..61 x 69

So last 2 digit of 69^33 = 09.

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Case 3 : The unit’s digit of a is 2,4,6 or 8.

If a ends in 2, 4, 6, 0r 8, we can find the last two digits of the number raised to power keeping in mind the following points :

2^10 ends in 24.

24^odd number ends in 24.

24^even number ends in 76.

76^number ends in 76.

For e.g. : **Find last 2 digits of number 2^2046 ?**

2^2046 = (2^10)^204 x 2^6 =(..24)^204 x 2^6

= (..76) x 64

Last 2 digit of the number 2^2046 = ..64

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Case 4 : The unit’s digit of a is 5.

If the digit in the tens place is **odd** and the exponent b is **odd**, then the number ends in 75.

If the digit in the tens place is **odd** and the exponent b is **even**, then the number ends in 25.

If the digit in the tens place is **even** and the exponent b is **odd**, then the number ends in 25.

If the digit in the tens place is **even** and the exponent b is **even**, then the number ends in 25 .

For e.g. : **Find last 2 digits of number 55^246 ?**

last 2 digit of 55^246 = 25

For e.g. : **Find last 2 digits of number 55^245 ?**

last 2 digit of 55^246 = 75.

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*Questions for practice :*

*Find last 2 digits of product 46^345?**Find last 2 digits of product 45 ^456 ?**Find last 2 digits of product 44^58 ?*