Problem Solving

what is the least positive integer that is divisible by each of the integers 1 thorough 7, inclusive?

a. 420
b. 840
c. 1260
d. 2520
e. 5040

answer is A.

Could someone please explain this? I am having hard time understanding OG's explanation? Thanks

ST

It's basically asking which of the smaller numbers given can be divided by 1,2,3,4,5,6 and7.
Get the first one, 420. obviously it can be divided by 1. we know it can be divided by 2 b/c its an even number. By 3 b/c all the numbers add up in a multiple of 3 and so on.....
If you are not 100% sure, divide individually the numbers...
Hope it helps...

Thank you. Make sense. It would be nice to learn rephrase technique also.

ST

If you break all numbers from 1 to 7 into prime factors, you get.

1 = 1^1
2 = 2^1
3 = 3^1
4 = 2^2
5 = 5^1
6 = 2^1 * 3^1
7 = 7^1

So the minimal prime factors that the least positive integer must have is 1^1 * 2^2 * 3^1* 5^1 *7^1 = 1*4*3*5*7 = 420.

If on the test, you can start checking the answers from the smallest number since the calculations are fairly simple.

from my point of view they have asked you to find the least common multiple.
L.C.M of 1,2,3,4,5,6,7 which is 420

ST wrote:what is the least positive integer that is divisible by each of the integers 1 thorough 7, inclusive?

a. 420
b. 840
c. 1260
d. 2520
e. 5040

Could someone please explain this? I am having hard time understanding OG's explanation? Thanks

ST

We're asked to find the lowest common multiple ("LCM") of the integers 1 through 7.

We know that 7! = 5040 will be a multiple of all 7, but that doesn't mean it's the lowest common multiple. The only time that the LCM and the multiple of a set of numbers is the same is when those numbers have no common prime factors.

There are many approaches we could take to find the LCM; as usual, some basic math and common sense/logic provides an efficient approach.

Let's take a quick peek at the set:

{1, 2, 3, 4, 5, 6, 7}

All of our primes must be included... so we're going to need a 2, 3, 5 and 7 among our factors.

1 is just 1 - we can ignore it.

4 = 2*2. We have one 2 already, so we need a second 2 to ensure that 4 is a factor of our number.

6 = 2*3. Since we already have a 2 and a 3, we can ignore 6.

Therefore, the LCM is 2*3*5*7*2 = 420.