Is my understanding right on the divisiblity rule?

Question

For a given number and a divisor. If the prime factors of the divisor can divide a number,then can I say that the divisor will divide a number.

For example - 786

divide by 21

If I break 21 in the prime factors - 3 * 7.

So, if the number is divisible by 3 as well as 7 that means the number can be divided by 21. Here 786 is not divisible by 21 because 7 cannot divide it completely.

Similarly, 42 will be broken as - 3 * 7 *2 -- this means it will not be divisible again. Since 7 couldn't divide it.

Is it a right method?

Answer 1

Yes, that's right. It is a consequence of the fact that every integer can be written as a product of powers of primes in an essentially unique way (the only lack of uniqueness is in the order in which you write down the factors).

So in your example, $-786 = -2\cdot 3\cdot 131$ (and $131$ is prime), and $21 = 3\cdot 7$. So if $21$ divided $786$, the primes in its prime factorization would appear in the prime factorization of $786$ by uniqueness of factorization. But $7$ does not, so that $7$, and thus $21$, do not divide $-786$. And since $21\nmid 786$, it follows that $2\cdot 21 = 42$ does not divide $786$ either.