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Which of the following CANNOT be the least common multiple

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by Brent@GMATPrepNow » Thu Feb 22, 2018 7:32 am
lheiannie07 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
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

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

Answer: D

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by Brent@GMATPrepNow » Thu Feb 22, 2018 8:35 am
lheiannie07 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
I was going to use some values for x and y to ELIMINATE A, B, C and E, but then realized that E is also correct (unless I'm missing something obvious).

If we CAN find values for x and y in which an answer choice IS the LCM of x and y, then we can ELIMINATE that answer choice.

(A) xy. If x = 2 and y = 3, then xy = 6, which means xy IS the LCM of x and y. ELIMINATE A

(B) x. If x = 4 and y = 2, then x IS the LCM of x and y. ELIMINATE B

(C) y. If x = 2 and y = 4, then y IS the LCM of x and y. ELIMINATE C

(E) x + y. Hmmmmmm. Can anyone find numbers so that x+y IS the LCM of x and y?

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by Jeff@TargetTestPrep » Mon Feb 26, 2018 10:00 am
lheiannie07 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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