Why do we check up to the square root of a prime number to determine if it is prime?

Question

To test whether a number is prime or not, why do we have to test whether it is divisible only up to the square root of that number?

Answer 1

A more intuitive explanation would be :-

The square root of 100 is 10. Let's say a x b = 100, for various pairs of a and b.

If a == b, then they are equal, and are the square root of 100, exactly. Which is 10.

If one of them is less than 10, the other has to be greater. For example, 5 x 20 == 100. One is greater than 10, the other is less than 10.

Thinking about a x b, if one of them goes down, the other must get bigger to compensate, so the product stays at 100. They pivot around the square root.

The square root of 101 is about 10.049875621. So if you're testing the number 101 for primality, you only need to try the integers up through 10, including 10. But 8, 9, and 10 are not themselves prime, so you only have to test up through 7, which is prime.

Because if there's a pair of factors with one of the numbers bigger than 10, the other of the pair has to be less than 10. If the smaller one doesn't exist, there is no matching larger factor of 101.

If you're testing 121, the square root is 11. You have to test the prime integers 1 through 11 (inclusive) to see if it goes in evenly. 11 goes in 11 times, so 121 is not prime. If you had stopped at 10, and not tested 11, you would have missed 11.

You have to test every prime integer greater than 2, but less than or equal to the square root, assuming you are only testing odd numbers.

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