(Number Theory) Verify that 281 is a prime number by completing the following co

Question

(Number Theory) Verify that 281 is a prime number by completing the following computations:

The remainder when 281 is divided by 2: 1

The remainder when 281 is divided by 3: 2

The remainder when 281 is divided by 5:1

The remainder when 281 is divided by 7:1

The remainder when 281 is divided by 11:6

The remainder when 281 is divided by 13: 8

Explain why the above computations are enough to show that 281 is prime.

I got the remainder part, I just don't know how to use that to show that 281 is prime.

Explanation / Answer

n = 281

sqrt(281) = 16.763

the biggest prime number less than sqrt(n) is 13

since none of the prime less than or equal to the biggest prime number less than sqrt(n) is able to divide n completety

we conclude that n is prime

in other words

If 281 were not a prime number, then it would be divisible by at least one prime number less than or equal to 281 16.763. Since 281 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 281 is a prime number.

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