If the least common multiple of integers x and y is 840,

Question

If the least common multiple of integers x and y is 840, what is the value of x?

(1) The greatest common factor of x and y is 56. 
(2) y = 168

KarishmaB · Answer

Remember a property of LCM and GCF:
If x and y are two positive integers, LCM * GCF = x* y 
Stmnt 1: Just the GCF will give you the product of the two numbers. Not their individual values. 
Stmnt 2: Knowing the value of y and LCM is not sufficient to get x.
x could be 840 or 105 or 5 etc.

Using both together, you get x = 840*56/168 = 280
Answer (C).

gettinit · Answer

Karishma is this a known property? I have never ran across it before? I am not sure I understand why it works either. Can you please explain?

KarishmaB · Answer

It is taught at school (though curriculums across the world vary)

Let us take an example to see why this works:

x = 60 = 2^2 * 3 * 5 
 y = 126 = 2*3^2*7

Now GCF here will be  6 (= 2*3)  (because that is all that is common to x and y)
LCM will be whatever is common taken once and the remaining i.e.  (2*3) * 2*5 * 3*7

When you multiply GCF with LCM, you get
 (2*3) * (2*3 *2*5 * 3*7)  i.e. whatever is common comes twice and everything else that the two numbers had.

I can re-arrange this product to write it as  (2*3 * 2*5) * (2*3 * 3*7)  i.e. 60*126

This is the product of the two numbers.
Since GCF has what is common to them and LCM has what is common to them written once and everything else, GCF and LCM will always multiply to give the two numbers.

Can you now think what will happen in case of 3 numbers?

Bunuel · Answer

The property Karishma used is often tested on GMAT. So, it's a must know property: for any positive integers x and y , x*y=GCD(x,y)*LCM(x,y) . See GMAT questions about this concept: xy-a-multiple-of-102540.html?hilit=most%20important#p797667 data-sufficiency-problem-95872.html?hilit=most%20important#p737970 Hope it helps.

gettinit · Answer

Thank you Karishma, makes sense. If I use three numbers I don't think this property will work based on the following example:

36 - 2^2*3^2
90 - 2*5*3^2
72- 2^3 * 3^2

GCF - 2*3^2 = 18
LCM - 2^3*3^2*5 = 360

so gcf*lcm=360*18=6480 which does not equal 36*90*72.

Bunuel thanks for the examples, helpful in reinforcing.

KarishmaB · Answer

Yes, that's right. It works only for two numbers.

shan123 · Answer

Karishma,

Thanks for the detailed explanation.

Shanif.
 Ur imagination is ur only limit

manwiththeharmonica · Answer

The question does not say the integers are positive. Is it implied? Factors are always positive so are GCDs and LCMs, but x could be a positive or a negative integer?

Posted from my mobile device

manwiththeharmonica · Answer

In that case the answer is E

Posted from my mobile device

AccipiterQ · Answer

You've got to be kidding me....that makes perfect sense. I'm glad I'm taking the GMAT, I'm learning all these fascinating formulas I've never used before.

AccipiterQ · Answer

Is there a sheet with the 'must-know' type stuff on it? I know there's the PDF you guys put together, but is there anything that is just a one or two page listing of the commonly used formulas like this?

KarishmaB · Answer

The list of must-know formulas would be really short and you would know most of the formulas on it (e.g. Distance = Speed*Time, Sum of first n positive integers = n(n+1)/2, area of a circle = pi*r^2 etc). Even if there are a couple that you don't know, you will come across them while preparing so just jot them down.

There will be many more formulas that you could find useful in particular questions but you can very easily manage without them. Also, learning too many formulas creates confusion about their usage - when to use which one - and hence they should be avoided.

ronr34 · Answer

Hi Karishma,

Stmnt 2: if we know the value of y, and the least common multiple, i can't think of another number that X could be other than 5.
Can you give an example?

KarishmaB · Answer

I have given some values that x can take from statement 2 - "x could be 840 or 105 or 5 etc"
Note that we don't know the value of GCF from stmnt 2 alone. Hence, x could take many different values. Only when we know the GCF too, can we find the value of x.

raj4ueclerx · Answer

Hi Karishma:

Can you please help me understand how can one find other numbers of x whose LCM will 840?

Harley1980 · Answer

Hello raj4ueclerx . Here is nice thread exclusively about finding LCM and GCF having-issues-with-finding-lcm-and-gcf-can-someone-help-146965.html

raj4ueclerx · Answer

Thanks for the suggestion....but I am still unable to find...How to come up with various values of X given that LCM(x, Y=168) = 840. Can some one please help me or show me the reverse calculation to derive x?

Harley1980 · Answer

Hello  raj4ueclerx

at first you should find all primes in both numbers
168 = 2*2*2*3*7
840 = 2*2*2*3*5*7

So we see that 840 has 5 as prime and 168 not. This number is only difference between 168 and 840
so any number that equal to product of this numbers 2, 2, 2, 3, 5, 7 (any combination that include 5) will give as LCM = 840 with number 168
for example 
2*5 = 10 LCM (10, 168) = 840 
3*5 = 15 LCM (15, 168) = 840 
2*2*2*5 = 40 LCM (40, 168) = 840 
and so on

Does that makes sense?

raj4ueclerx · Answer

at first you should find all primes in both numbers
168 = 2*2*2*3*7
840 = 2*2*2*3*5*7

So we see that 840 has 5 as prime and 168 not. This number is only difference between 168 and 840
so any number that equal to product of this numbers 2, 2, 2, 3, 5, 7 (any combination that include 5) will give as LCM = 840 with number 168
for example
2*5 = 10 LCM (10, 168) = 840
3*5 = 15 LCM (15, 168) = 840
2*2*2*5 = 40 LCM (40, 168) = 840
and so on

Does that makes sense?[/quote]

Million thanks ...this helps

If the least common multiple of integers x and y is 840,

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