MHB Solving 7x=210: Must x be a Multiple of 12?

[ChickenWings]

Please help me understand how to get the solution to the following:

If 7x is a multiple of 210, must x be a multiple of 12?

Thank you!

 

[MarkFL]

I would begin by finding the prime factorization of 210, which is:

$$210=2\cdot3\cdot5\cdot7$$

Now, if $7x$ is a multiple of 210, then we may write:

$$7x=210n=2\cdot3\cdot5\cdot7\cdot n$$

Now dividing through by 7, we find:

$$x=2\cdot3\cdot5\cdot n$$

What can you conclude from this?

 

[ChickenWings]

I really appreciate your response but I'm getting lost at the "$$7x=210n=2\cdot3\cdot5\cdot7\cdot n$$" mark. Sorry. :(

I would begin by finding the prime factorization of 210, which is:

$$210=2\cdot3\cdot5\cdot7$$

Now, if $7x$ is a multiple of 210, then we may write:

$$7x=210n=2\cdot3\cdot5\cdot7\cdot n$$

Now dividing through by 7, we find:

$$x=2\cdot3\cdot5\cdot n$$

What can you conclude from this?

 

[MarkFL]

If $n$ is a natural or counting number, then the multiples of 210 can be written as $210n$. All we did is then turn the statement "7x is a multiple of 210" into an equation. :D