What is the lowest positive integer that is divisble by each of the integers 1 through 7, inclusive?

What's the best way to do this? The solution tells you to use the lowest common multiple of 2,3,4 and 6. What word in the question makes you recognize this is the proper way to do the problem?

Several words are important here.

First, if an integer is divisible by certain other integers, those other integers are factors of the main integer in question, and I can create the integer in question by multiplying those factors.

So if I want an integer (but not necessarily the smallest) that is divisible by 1 thru 7 inclusive, I need something that has all the necessary components as factors. That means I need to multiply: 1*2*3*4*5*6*7.

The question also says, though, that it wants the LOWEST integer that fits the criteria, not just any integer. So I want to use the minimum necessary when figuring out what to multiply:
1
2 (have I included a 2 yet? No, so keep this one in)
3 (have I included a 3 yet? No, so keep this one in)
4, or 2*2 (have I included any 2's yet? Yes, I've included one from above. How many more do I need to make a 4? Just one more 2. So add in only one 2 here, not two 2's)
5 (have I included a 5 yet? No, so keep this one in)
6, or 2*3 (have I included any 2's or 3's yet? Yes, I've got a 2 up above and a 3 up above, so I've already created my 6. I don't need to include anything more here)
7 (have I included a 7 yet? No, so keep this one in)

That leaves me with: 1*2*3*2*5*7.
(And, of course, you can ignore the 1, since it won't change the answer.)

Is there a quicker way to do that? After reading the question text should you recognize this is an LCM question?

Yes, I was laying it out for you step by step to make sure you understood - the culmination is that it is an LCM problem. So you wouldn't go through the first set of things I wrote down. You'd just recognize that you want the minimum necessary, based on the words "lowest" and "integer that is divisible by" and write down: 2*3*2*5*7 (asking yourself that series of questions I typed for each number).

Thank you Stacey. Your explanation for LCM helped me understand the concept a little better.

Great!