What is the largest number that has exactly 7 factors?

Answer 1

There is no largest number of that sort, as there is an infinite quantity of numbers with seven factors.

Here's why:

To get a number with exactly seven factors, all you need to do is take seven prime numbers and multiply them together. For example, 2 * 3 * 5 * 7 * 11 * 13 * 17 = 510510. So to get the largest possible number with exactly seven factors, you would need to find the seven largest prime numbers. This is where the problem lies. There are an infinite number of primes, and thus no largest one.

One commonly used reasoning of this is as follows: Imagine collecting all of the prime numbers up to a certain amount. If you multiply them all together and add one, the resulting number will not be divisible by any of them, and must therefore be prime or have prime factors that are not in your list. Therefore you could not have known them all, and by that logic, never can.

For example, we know that the numbers two, three, and five are prime. Let's multiply them together and add one. 2 * 3 * 5 = 30, and 30 + 1 gives us 31, another Prime number.

Knowing then that there are an infinite number of primes, we can similarly say that there are an infinite number of values with exactly seven factors, and therefore no largest one.