If the greatest common factor of two integers, m and n, is 56 and the

Question

If the greatest common factor of two integers, m and n, is 56 and the least common multiple is 840, what is the sum of the m and n? 
 
(1) m is not divisible by 15.
(2) n is divisible by 15.

LHC8717 · Answer

Answer D

From the statement GCF : 56 = 2^3 * 7 LCM : 840 = 2^3 * 3 * 5 * 7

Therefore m and b should have 2^3 * 7 as factor and the other missing are 3 * 5 to complete the LCM

(1) m has no factor 3 and 5 => n should have 3 and 5 SUFFICIENT
(2) n has 3 and 5 as factor => m could not have any other prime number otherwise the GCF and LCM will change SUFFICIENT

sillyboy · Answer

B.

GCD(m,n) = 56
LCM(m,n) = 840

Let m = Gx and n = Gy
Where G = GCD(m,n) and (x,y) are coprimes

Now, m*n = (Gx)(Gy) = GCD * LCM
xyG^2 = 56*840
xy = (56*840)/(56^2) = 15

(factors of 15 = 1,3,5,15)
Therefore, possible solutions for (m,n) are (56*1 , 56*15) or (56*15 , 56*1) AND (56*3 , 56*5) or (56*5 , 56*3) => 4 cases

Now,
S1: m is not divisible by 15.
=> only case 1, 3, and 4 satisfy => NOT SUFFICIENT

S2: n is divisible by 15
=> only case 1 satisfy => SUFFICIENT

RMD007 · Answer

GCF = 56 = 2^3*7 ==> Both M and N have 2^3 and 7 as factors
LCM = 840 = 2^3*7*3*5

Statement 1 : if M is not divisible by 15, the possible combinations can be
M= 2^3*7 N= 2^3*7*3*5
M=2^3*7*3 N = 2^3*7*5
N=2^3*7*5 N = 2^3*7*3

Not Sufficient.

Statement 2: ==> N = 2^3*7*5*3 which means M= 2^3*7 only one combination. ==> Sufficient.

B.

LHC8717 · Answer

Oh right, I did a silly mistake to not consider your second and third cases for the first statement.

musunna · Answer

Excellent explanation. Thanks.

But i think there is a typo in the quoted portion.

Shouldn't it be M= 2^3*7   N= 2^3*7*3*5

RMD007 · Answer

Thanks, I corrected!

Deep32470 · Answer

If the greatest common factor of two integers, m and n, is 56 and the least common multiple is 840, what is the sum of the m and n?

(1) m is not divisible by 15.
(2) n is divisible by 15.

I have reached the conclusion with different approach.
If we know GCF & LCM of two nos then the product of those two numbers will be equal to the product of GCF & LCM.
Hence, m*n=47040=2^6*7^2*3*5

Using option (1), m can be 10, 6 or anything from above & n can be different & hence sum of a+b would be different with this condition in each cases.

Using option (2), also we can have n=15, 30, 90 and so on and hence m can be according to the value of n & hence m+n would be different in each cases.

Now if we combine both, then if

n=15, then m=3136,
and if n=30, then m=1568 and sum of m+n would be different in above case too.
So in my opinion, E should be the option.

bumpbot · Answer

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