Why would you get the LCM of the given numbers when you multiply the highest power of the prime factors?

Answer 1

This is the reverse of your other question.

You want to keep the number as small as possible, but you have to guarantee that all the numbers divide into it.

So if you take the highest power of each prime factor, you're guaranteeing that every one of the numbers you start with will divide into the LCM...but you're avoiding duplication.

Another way to think about it, again with 16 and 12:

16 = 2 × 2 × 2 × 2
12 = 2 × 2 × 3

So the LCM is 2 × 2 × 2 × 2 × 3, or 48.

This is the same as multiplying 16 (2 × 2 × 2 × 2) by 12 (2 × 2 × 3) and getting 192...and then dividing by the "overage" of 2 × 2 that you would have eliminated by only using the highest power of each of the numbers.

As a non-numeric example...suppose that the 3 guys from my last example are diplomats representing their individual countries. The French diplomat is only willing to send 0-500 NATO soldiers to Iraq, but is ok with 0-1,000 in Afghanistan. The Spanish diplomat is willing to send 0-2,000 to Iraq, but only 0-500 to Afghanistan. The American diplomat hopes for 500-2,000 soldiers to each.

The least number of soldiers that could be sent that wouldn't cause an argument about sending too many would be 500 in Iraq and 500 in Afghanistan; note that these are the largest amounts that all three negotiators have in mind as being acceptable.

Answer 2

Multiplying the highest power of the prime factors gives you a number which each of the given numbers will divide into evenly.

Reducing any of the powers, even by 1, gives you a smaller number, which at least one of the given numbers (the one with that power before you reduced it) will no longer divide evenly into.

So the first number is the LEAST common multiple. Nothing smaller will do.