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If mx = m, then what is the value of m?

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Fri Mar 09, 2018 12:18 am
VJesus12 wrote:If mx = m, then what is the value of m?

(1) m has only one multiple.
(2) x has only one distinct factor.

The OA is A.

Why is not sufficient the statement (2)? Experts, may you help me to solve this DS question? Thanks in advanced.
Given: mx = m

We have to find out the value of m.

Let's take each statement one by one.

(1) m has only one multiple.

0 is the only number that has only one multiple, thus m = 0. Sufficient. Sufficient. Note multiples of 1 are 1, 2, 3, ...

(2) x has only one distinct factor.

1 is the only number that has only one distinct factor, thus x = 1, but we cannot get the unique value m. m can take any value; for example 0*1 = 0; 10*1 = 10, etc. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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by Vincen » Fri Mar 09, 2018 7:27 am
Hello Vjesus12.

I'd solve it as follows:

(1) m has only one multiple.

This implies that, given any integers x and y, we get that $$m\cdot x=m\cdot y\ \ \ \ \forall\ x,y\ \in R.$$ Now, the only number that satisfy this condition is m=0.

Hence, this statement is Sufficient.

(2) x has only one factor.

This implies that the prime factorization of x is only one number, and the unique number that satisfy this condition is x=1.

Therefore, we have that mx=m implies that m=m. But this doesn't tell us what is the value of m. INSUFFICIENT.

Because of this, the answer is the option A.
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