Question

How can you use prime factorization to determine if 738 is evenly divisible by 14?

Answer 1

See explanation.

Explanation:

It is difficult to divide `738` by `14` without a calculator, but it is easy to see, that:

`14=2*7`

So if `738` was to divide by `14` it would have to divide by both `2` and `7`

It is an even number, so it divides by `2`, but it does not divide by `7`. We can see this if we write

`738=700+38`

`700` clearly divides by `7` but `38` does not, because it lies between 2 consecutive multiples of `7`: `35<38<42`.

So because `738` does not divide by `7` it cannot divide by `14`.

This example shows that to check if one number divides by the other you can find the prime factorization of the second number and check if the first number divides by all prime factors of the second one.