Question:
If $5^{97}$ is divided by $52$ $,$ then the remainder obtained is
If $5^{97}$ is divided by $52$ $,$ then the remainder obtained is
Updated On: Jul 28, 2022
- $3$
- $5$
- $4$
- $0$
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The Correct Option is B
Solution and Explanation
We know that, $5^{4}=625=52\times 12+1$
$\Rightarrow 5^{4}=52\lambda +1$ , where $\lambda $ is a positive integer.
$\Rightarrow \left(5^{4}\right)^{24}=\left(52 \lambda + 1\right)^{24}$
$= \, \_{}^{24}C_{0}\left(52 \lambda \right)^{24}+\_{}^{24}\left(C_{1} \left(52 \lambda \right)^{23} + \_{}^{24}C_{2} \left(52 \lambda \right)^{22}\right)+\ldots +\_{}^{24}\left(C_{23} \left(52 \lambda \right) + \_{}^{24}C_{24}\right)$ (by binomial theorem)
$\Rightarrow 5^{96}=52\left[\_{}^{24}C_{0} 52^{23} \lambda ^{24} + \_{}^{24}C_{1} 52^{23} \lambda ^{22} + \ldots + \_{}^{24}C_{23} \lambda \right]+1$
$=\left(a \, m u l t i p l e \, o f \, 52\right)+1$
On multiplying both sides by $5$ , we get
$5^{97}=5^{96}\cdot 5=5 \, \left(a \, m u l t i p l e \, o f \, 52\right)+5$
Hence, the required remainder is $5$ .
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.