Smallest Common Multiple

Stuart Kovinsky wrote:

deepesh.gupta wrote:What is the smallest common multiple of two integers which are both greater than 250?
1) 251
2) 252
3) 502
4) 750
5) 884

Hi,

we have to be very careful about not making assumptions on the GMAT; the GMAT uses very precise language and we should never "read in" a word that isn't actually there.

One commonly tested case involves the word "distinct", which simply means "different" - you have to be very careful not to add that word to a question.

This question is a perfect example - nowhere does it say that the two integers have to be distinct integers.

Accordingly, we can pick 251 and 251 as our two integers, which have a lowest common multiple of 251: choose (1).

Hello Sir, Why is 251 the smallest multiple? It should be zero. Please guide.

Hello Brent,

Thanks!
So you agree that the answer is incorrect? It should be zero.

Regards,
Zee

This is an apropos discussion, as there's even an official GMAT question, available in the new Exam Pack, that seems to me to make this same mistake. I asked the GMAC rep on BTG for some insight into their thinking, but she hasn't responded yet. (The thread is here. The official answer is E, but the actual answer is A.)

My take on all this: the Least Common Multiple (LCM) is explicitly defined as the least POSITIVE common multiple of x and y. Without that definition (or some equivalent), there is no smallest common multiple, since we can pick an arbitrarily small negative common multiple. (For instance, -6 is a multiple of 3 and 2, but so is -12, and -24, etc.) So a term like "smallest common multiple" is self-contradicting, unless it's the smallest common multiple of 0 and 0, which wouldn't be terribly interesting!

0 is not multiple of an integer.


Please suggest.

251 is the right answer.

HI Brent

If 0 is multiple of any integer then

LCM of 4 & 5 should be 0 not 20.

Please suggest.

nikhilgmat31 wrote:HI Brent

If 0 is multiple of any integer then

LCM of 4 & 5 should be 0 not 20.

Please suggest.

I addressed this about a few posts earlier, check the (long) explanation there.

The short answer is that LCM is defined as the least common POSITIVE multiple.

Otherwise there would not be an LCM, since you could take arbitrarily small negative common multiples.

Thanks Brent, It answers my question.

deepesh.gupta wrote:What is the smallest common multiple of two DISTINCT integers which are both greater than 250?
1) 251
2) 252
3) 502
4) 750
5) 884

P.S. Have made a small modification in question to make it a little better

First Number greater than 250 = 251

Second Number with maximum factor of it common with first number = 502 (smallest integer multiple of 251 greater than 251)

LCM = 502

Answer: Option C

Got it

5*5*5*2 = 250
Smallest common multiple = 250*2, 250*3, 250*4
So the answer is 4