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.
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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 ?
