# Find Remainder When Divided by 19

• MHB
• Deanmark
In summary, when computing the remainder of 2^(2^17) + 1 when divided by 19, we first compute 2^17 mod 18 because of the Little Theorem of Fermat. This allows us to simplify the expression and find the remainder more easily.
Deanmark
Compute the remainder of 2^(2^17) + 1 when divided by 19. The book says to first compute 2^17 mod 18 but I don’t understand why we go to mod 18. Advice would be appreciated

Deanmark said:
Compute the remainder of 2^(2^17) + 1 when divided by 19. The book says to first compute 2^17 mod 18 but I don’t understand why we go to mod 18. Advice would be appreciated

Hi Deanmark,

That's because of the Little Theorem of Fermat:
$$a^{p-1} \bmod p = 1$$
where $p$ is prime and $a$ is any number except for a multiple of $p$.

So if we can write $2^{17}$ as some multiple of $18$ and a remainder, say $2^{17} = 18k + r$, then:
$$2^{(2^{17})} + 1 \bmod 19 = 2^{18k+r} + 1 \bmod 19 = (2^{18})^k\cdot 2^r + 1 \bmod 19 = 2^r + 1 \bmod 19$$

The parts of my wall that have yet to be punched thank you.

## What is the purpose of finding the remainder when divided by 19?

The purpose of finding the remainder when divided by 19 is to determine the value that is left over after dividing a number by 19. This can be useful in various mathematical calculations, such as modular arithmetic or finding the smallest period of a repeating decimal.

## How do you find the remainder when divided by 19?

To find the remainder when divided by 19, you can use the modulo operator (%) in most programming languages or use long division by hand. The modulo operator returns the remainder after dividing two numbers, while long division involves repeatedly subtracting multiples of 19 from the original number until you reach a remainder of less than 19.

## What is the significance of 19 in finding the remainder?

The number 19 is significant in finding the remainder because it is the divisor in the given problem. The remainder when dividing by 19 can range from 0 to 18, making it a useful number in various mathematical calculations.

## Why is finding the remainder when divided by 19 important in computer science?

In computer science, finding the remainder when divided by 19 is important because it is used in many algorithms and data structures. For example, it can be used in hash functions, where the remainder is taken after dividing by a prime number (such as 19) to reduce the likelihood of collisions.

## Can finding the remainder when divided by 19 be used to solve real-world problems?

Yes, finding the remainder when divided by 19 can be used to solve real-world problems. For example, it can be used in financial calculations, such as calculating interest rates or determining the number of payments needed to pay off a loan. It can also be used in physics and engineering, such as determining the period of a repeating waveform or the number of rotations in a mechanical system.

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