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Which of the following CANNOT be the least common multiple

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by BTGmoderatorDC » Sun Aug 05, 2018 4:18 am
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Which of the following CANNOT be the least common multiple of two positive integers x and y?

(A) xy
(B) x
(C) y
(D) x - y
(E) x + y

OA D

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by Brent@GMATPrepNow » Sun Aug 05, 2018 4:56 am
BTGmoderatorDC wrote:Which of the following CANNOT be the least common multiple of two positive integers x and y?

(A) xy
(B) x
(C) y
(D) x - y
(E) x + y

OA D

Source: GMAT Prep
Key Concept: the least common multiple (LCM) of x and y is a multiple of x and a multiple of y.

Let's examine a few multiples:
Multiples of 7: 7, 14, 21, 28, 35,...
Multiples of 10: 10, 20, 30, 40, 50, 60, ,...
Multiples of 3: 3, 6, 9, 12, 15, 18,,...
Notice that the multiples of N are always greater than or equal to N
So, the LCM of x and y must be greater than or equal to x AND greater than or equal to y

Answer choice D (x - y), suggests that x-y is a multiple of x and y
HOWEVER, we can see that x-y is clearly LESS THAN x
As such, x-y cannot be a multiple of x, which also means x-y cannot be the LCM of x and y

Answer: D

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Brent
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LCM

by GMATGuruNY » Sun Aug 05, 2018 5:19 am
BTGmoderatorDC wrote:Which of the following CANNOT be the least common multiple of two positive integers x and y?

(A) xy
(B) x
(C) y
(D) x - y
(E) x + y
Neither x-y nor x+y can be the LCM of x and y.
What is the source of this problem?
Last edited by GMATGuruNY on Sun Aug 05, 2018 5:38 am, edited 1 time in total.
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by Brent@GMATPrepNow » Sun Aug 05, 2018 5:24 am
BTGmoderatorDC wrote:Which of the following CANNOT be the least common multiple of two positive integers x and y?

(A) xy
(B) x
(C) y
(D) x - y
(E) x + y
Am I going crazy?
I was also going to solve this question using the process of elimination.
That is, if an answer choice CAN be the least common multiple (LCM) of x and y, then we'll ELIMINATE that answer choice.

(A) If x = 1 and y = 1, then the LCM is 1. Since xy = (1)(1) = 1, we can see that xy CAN be the LCM of x and y. ELIMINATE A

(B) If x = 1 and y = 1, then the LCM is 1. So x CAN be the LCM of x and y. ELIMINATE B

(C) If x = 1 and y = 1, then the LCM is 1. So y CAN be the LCM of x and y. ELIMINATE C

(E) x + y
For the life of me, I cannot think of values of x and y such that the LCM is x+y.
Help!

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by GMATGuruNY » Mon Aug 06, 2018 5:41 am
Brent@GMATPrepNow wrote:(E) x + y
For the life of me, I cannot think of values of x and y such that the LCM is x+y.
Help!

Cheers,
Brent
If x and y are equal, then their LCM = x = y.
Thus, x+y can be the LCM of x and y only if x and y are NOT equal.
If x+y is the LCM of x and y, then dividing x+y by x or y must yield an INTEGER.
If we divide x+y by x. we get:
(x+y)/x = x/x + y/x = 1 + y/x.
The expression in blue will be an integer only if y > x.
If we divide x+y by y, we get:
(x+y)/y = x/y + y/y = x/y + 1.
If the expression in blue is an integer -- implying that y > x -- then the expression in red CANNOT be an integer, since x/y = (smaller)/(bigger) = fraction.
Since it is not possible for x+y to be divisible by both x and y, option E cannot be the LCM of x and y.
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by Brent@GMATPrepNow » Mon Aug 06, 2018 6:56 am
GMATGuruNY wrote:
Brent@GMATPrepNow wrote:(E) x + y
For the life of me, I cannot think of values of x and y such that the LCM is x+y.
Help!

Cheers,
Brent
If x and y are equal, then their LCM = x = y.
Thus, x+y can be the LCM of x and y only if x and y are NOT equal.
If x+y is the LCM of x and y, then dividing x+y by x or y must yield an INTEGER.
If we divide x+y by x. we get:
(x+y)/x = x/x + y/x = 1 + y/x.
The expression in blue will be an integer only if y > x.
If we divide x+y by y, we get:
(x+y)/y = x/y + y/y = x/y + 1.
If the expression in blue is an integer -- implying that y > x -- then the expression in red CANNOT be an integer, since x/y = (smaller)/(bigger) = fraction.
Since it is not possible for x+y to be divisible by both x and y, option E cannot be the LCM of x and y.
Phew! I thought I was going bananas!
Obviously not an actual GMATPrep question (as the OP suggests). I wonder what the original answer choice E was.
I searched for this post elsewhere, and it appears to have been transcribed like that each time.

Cheers,
Brent
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by Jeff@TargetTestPrep » Sat Aug 18, 2018 6:57 pm
BTGmoderatorDC wrote:Which of the following CANNOT be the least common multiple of two positive integers x and y?

(A) xy
(B) x
(C) y
(D) x - y
(E) x + y
Since the difference of x - y is less than x, the quantity x - y can’t be a multiple of x. Thus, it can’t be the least common multiple (LCM) of x and y.

(Note: We think the correct answer is intended to be D for the reason stated above, but choice E is also correct since x + y can’t be the LCM of x and y, either. We can prove this by contradiction:

Let’s suppose that x + y is the LCM of x and y. We see that x and y can’t be equal, otherwise either x or y (not their sum) will be the LCM of x and y. Now let’s say that x < y. Since we suppose that x + y is the LCM of x and y, y, the larger of the two numbers, can’t be the LCM of x and y. But the LCM of x and y must be the a multiple of y, so it has to be at least 2y (if it can’t be y). Here is the contradiction: 2y > x + y since y > x. So it’s impossible to have x + y as the LCM of x and y.)

Answer: D

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