Chapter 1: Q13E (page 48)

Is the difference of $5^{30, 000} and 6^{123, 456}$ a multiple of $31$ ?

Short Answer

Expert verified

The difference of $5^{30, 000} and 6^{123, 456}$ a multiple of 31 is 0, i.e. it is multiple of $31$ .

Step by step solution

Fermat’s Little Theorem

Fermat’s Little Theorem can be used as to calculate the power of the given integers with the help of modulo of prime numbers.

i.e.

$x^{n} - x$ is divisible by $n$ .

where,

$x =$ any integer

$n =$ prime number.

Calculation for 530,000

As $31$ is a prime number, that can be written as $30 + 1$ .

Then,

For $5^{30, 000}, 30000$ can be divisible by $30$ .

So , $5^{30, 000} \mod 31$ is 1.

Calculation for 6123,456

Here, $123456$ is not divisible by $30$ .

Then,

$123456 as 123450 + 6 and 123450 is divisible by 30 .$

So,

$\begin{matrix} 6^{123, 456} \mod 31 = 6^{123, 450} * 6^{6} (\mod 31) \\ = 6^{6} (\mod 31) \\ = 36 * 36 * 36 (\mod 31) \end{matrix}$

By dividing $36 by 31$ , the remainder will be $5$ .

Then,

$\begin{matrix} 5 * 5 * 5 \mod 31 = 125 \mod 31 \\ = 1 \end{matrix}$

Therefore, by subtracting the final answers of step $2$ and step $3$ we get the answer equal to $0$ . So, the difference of $5^{30, 000} and 6^{123, 456}$ a multiple of $31 is 0$ i.e. it is multiple of $31$ .

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Is the difference of $5^{30, 000} and 6^{123, 456}$ a multiple of $31$ ?

Short Answer

Step by step solution

Fermat’s Little Theorem

Calculation for 530,000

Calculation for 6123,456

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Is the difference of 530,000and6123,456a multiple of31 ?

Short Answer

Step by step solution

Fermat’s Little Theorem

Calculation for 530,000

Calculation for 6123,456

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Issues in Computer Science

Big Data

Game Design in Computer Science

Functional Programming

Computer Programming

Algorithms in Computer Science

Study anywhere. Anytime. Across all devices.

Is the difference of $5^{30, 000} and 6^{123, 456}$ a multiple of $31$ ?