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Re: When 51^25 is divided by 13
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01 Jul 2020, 13:03
1
Quick way to calculate: 51^1 / 13 --> remainder is 12 51^2 / 13 --> remainder is 1 51^3 / 13 --> remainder is 12
There is a cycle of 2 The power 25 is not divisible by 2 and has a remainder of 1. Therefore, the remainder of 51^25 / 13 is equal to the remainder of 51^1 / 13 --> 12
When 51^25 is divided by 13
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19 Oct 2022, 10:56
1
We need to find When \(51^{25}\) is divided by 13, the remainder obtained is
Now, to solve this problem we will be using the concept of Binomial Theorem
We will try to split 51 into two numbers. One number will be a multiple of 13 and will be very close to 51 and other number will be a very small number.
\(51^{25}\) = \((52 - 1)^{25}\) (52 = 13*4)
Now, when we open this using Binomial Theorem then all terms apart from the last term will be a multiple of 52 => a multiple of 13 => Remainder by 13 will be 0
=> Question is reduced to what is the remainder when the last term ( 25C25 \((-1)^{25} * 52^0\) is divided by 13
=> Remainder when -1 is divided by 13 => -1 + 13 = 12
So, Answer will be A Hope it helps!
Watch the following video to Learn about Binomial Theorem and How to Solve Similar problems