Problem Solving

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

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

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)

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.

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..

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.

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)