If the least common multiple

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figs: Senior | Next Rank: 100 Posts; Posts: 70; Joined: Mon Mar 30, 2009 9:00 am; Thanked: 2 times

by figs » Mon May 25, 2009 4:23 am

If the least common multiple between two positive integers is 120 and their ratio is 3 to 4, what is their greatest common divisor?

A. 5
B. 10
C. 12
D. 15
E. 20

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Source: Beat The GMAT — Problem Solving | Mon May 25, 2009 4:23 am

scoobydooby: Legendary Member; Posts: 1035; Joined: Wed Aug 27, 2008 10:56 pmFollowed by: 1 member; Thanked: 104 times

by scoobydooby » Mon May 25, 2009 5:47 am

let the numbers be 3x and 4x

LCM: 3x*4x/x (product of the numbers divided by the common factor)
or LCM=12x=120 (given)
x=10

or HCF is 10.
(check: product of the numbers: HCF*LCM)

hence, B

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ssmiles08: Master | Next Rank: 500 Posts; Posts: 472; Joined: Sun Mar 29, 2009 6:54 pm; Thanked: 56 times

by ssmiles08 » Mon May 25, 2009 6:11 am

120 can be broken down into primes = 2 * 2 * 3 * 5 * 2

The ratio of 3:4 has 3 * 2 * 2

The only thing is missing is 5*2. Since we have to preserve the ratio we have to multiply both sides by 10 to make it 30:40

Therefore the L.C.M of 30 and 40 is 120 and the H.C.F of 30 and 40 is 10 (B)

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dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

Re: If the least common multiple

by dtweah » Mon May 25, 2009 6:12 am

figs wrote:If the least common multiple between two positive integers is 120 and their ratio is 3 to 4, what is their greatest common divisor?

A. 5
B. 10
C. 12
D. 15
E. 20

120 =2^3 x 3 x 5

If you divide the above LCM by each of the numbers you will get the given ratios, since ratios are expressed in lowest terms. Ratio is given as is 3 to 4. If you get 3 then that means the divisor does not have 3 but has 2^3 and 5 as factors. If you get 4 then the divisor has a 2, has a 3 and has a 5 as factors. So we know 3 is not common to both.

the GCF follows as 2 x 5 which are the only common numbers. Choose B.

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dumb.doofus: Master | Next Rank: 500 Posts; Posts: 435; Joined: Sat Sep 27, 2008 2:02 pmFollowed by: 1 member; Location: San Jose, CA; Thanked: 43 times; GMAT Score:720

by dumb.doofus » Mon May 25, 2009 10:11 am

scoobydooby wrote:let the numbers be 3x and 4x

LCM: 3x*4x/x (product of the numbers divided by the common factor)
or LCM=12x=120 (given)
x=10

or HCF is 10.
(check: product of the numbers: HCF*LCM)

hence, B

Nice approach.. thanks for sharing..

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

Re: If the least common multiple

by Stuart@KaplanGMAT » Mon May 25, 2009 11:49 am

figs wrote:If the least common multiple between two positive integers is 120 and their ratio is 3 to 4, what is their greatest common divisor?

A. 5
B. 10
C. 12
D. 15
E. 20

Lots of great mathematical explanations for why (b) is correct. Understanding the math is important for GMAT success. However, it's also important to recognize cases in which alternative strategies, such as picking numbers, are quicker.

When I first read this question, the numbers 30 and 40 immediately jumped out at me. A quick double check shows that the LCM of 30 and 40 is indeed 120 and it doesn't take long to determine that the GCF of 30 and 40 is 10: choose (b).

What's the takeaway? (This question is the one that you should be asking after EVERY problem you tackle in practice.) In number property questions in which we're given some basic information about numbers but no actual values, picking numbers is an excellent way to shortcut the process.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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avanishjoshi: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Fri Jun 20, 2008 2:06 am; Thanked: 2 times

by avanishjoshi » Tue May 26, 2009 9:49 am

Another approach is as follows:

LCM * GCD = product of two numbers.

If the two numbers are 3x and 4x then

120 * GCD = 12X^2.

Now substitute the options.

(A) 5 ie 120 * 5 = 12X^2 => X^2 = 50 which is not possible as square of a natural number cannot be 50

(B) 10 ie 120*10 = 12X^2 => X^2 = 100 which is possible .

Other options do not yield a perfect number.

Therefore answer is 10 .. (B)

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