Divisibility and LCM

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metallicafan: Master | Next Rank: 500 Posts; Posts: 142; Joined: Sun Jun 12, 2011 7:55 amFollowed by: 3 members; Thanked: 5 times

by metallicafan » Wed Oct 17, 2012 11:23 am

Is the integer x divisible by 24?
(1) x is divisible by 8.
(2) x is divisible by 6.

The solution is easy. However, I have some conceptual doubts related to this problem.
To solve the problem we have to find the LCM, which is 24 in this question. Therefore, x is divisible by 24.

But what about the number 0. According to the MGMAT guide, 0 is a multiple of every integer. Let's see: "any integer * 0 = 0 "
In this sense, and based on that definition, the LCM wouldn't be 24 but it would be 0. Because 0 is a multiple of 8 and 6 and also is less than 24.
Why is 0 not considered the LCM of any set of integers?, Should we calculate the LCM or only analyze whether 0 is divisible by 24?

OA is C.

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Source: Beat The GMAT — Data Sufficiency | Wed Oct 17, 2012 11:23 am

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Whitney Garner: GMAT Instructor; Posts: 273; Joined: Tue Sep 21, 2010 5:37 amFollowed by: 74 members; Location: Durham, NC; Thanked: 154 times; GMAT Score:770

by Whitney Garner » Wed Oct 17, 2012 11:45 am

metallicafan wrote:Is the integer x divisible by 24?
(1) x is divisible by 8.
(2) x is divisible by 6.

The solution is easy. However, I have some conceptual doubts related to this problem.
To solve the problem we have to find the LCM, which is 24 in this question. Therefore, x is divisible by 24.

But what about the number 0. According to the MGMAT guide, 0 is a multiple of every integer. Let's see: "any integer * 0 = 0 "
In this sense, and based on that definition, the LCM wouldn't be 24 but it would be 0. Because 0 is a multiple of 8 and 6 and also is less than 24.
Why is 0 not considered the LCM of any set of integers?, Should we calculate the LCM or only analyze whether 0 is divisible by 24?

OA is C.

Hi metallicafan!

Great question about zero and you're right that 0 is the universal multiple (it is a multiple of every number, and is therefore divisible BY every number - but note that NOTHING is divisible BY zero!!)

Okay, so let's get a couple of definitions straight...

Common Multiple: a number that is a multiple of 2 or more numbers. So the common multiples of 6 and 8 are 0, 24, 48, 72, ... (notice that YES, 0 is a common multiple).

LEAST Common Multiple (LCM): or 2 (or more) numbers is the smallest NON-zero common multiple!

So by definition, the LCM cannot be 0. BUT we want to be careful with 0 at all. For example think about the following problem:

Is the integer y divisible by 27?
(1) y is divisible by 9
(2) y is divisible by 6

Remember that often our job is to try to PROVE insufficiency, so we are trying to show that we can get either a yes or a no depending on the numbers we pick. If we pick a number like 0 or 27 (both divisible by 9 and 6), we get an answer of YES to our problem (because both 0 and 27 are divisible by 27). BUT, if we pick the LCM, 18 (which is divisible by both 9 and 6). We get the answer NO to our problem (because 18 is NOT divisible by 27).

So the reason why we are even looking for the LCM isn't just because someone told us to. It is because we are trying to find a pattern regarding the possible "look" of the number in question. Since 0 is divisible by ALL numbers, knowing that 0 is divisible by our mystery number doesn't really help us. BUT, knowing the LCM and therefore the most simplistic (non-zero) structure of our number (if our number is divisible by 6 and 9, we know that our number is therefore a multiple of the LCM of 6 and 9).

I hope this helps explain why 0 is definitely a number to consider but why it adds little additional value!!

Whit

Whitney Garner
GMAT/GRE/EA Instructor & Anxiety/Accommodations Coach
www.whitneygarner.com

Contributor to Beat The GMAT!

Math is a lot like love - a simple idea that can easily get complicated

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