Is Zero a factor of Itself

Question

Hi, Can an expert let me know if zero is a factor of itself. If I am correct there are two ways to look at the definition of a factor and I am getting contradicting results by both (or I am terribly mistaken at the definition in itself - Either way, if the question sounds stupid, I apologize in advance

) Definition 1 If a is a factor of b then b / a is an integer. If I go by this definition, then 0 / 0 should be an integer, however anything divided by 0 is undefined. Hence, my "logical" (or not so logical

) brain says that zero is NOT a factor of itself. Definition 2 However, if I define factor as If a is a factor of b then - b= a * k (for some integer k) By this definition, 0 seem to be a factor of itself. In a nutshell, I am lost ! Can an expert help me out here and share his / her view on this. Tagging few quant wizards to the thread, however anyone is welcome to share an opinion - IanStewart chetan2u avigutman KarishmaB

avigutman · Answer

Zero is not a factor of itself (though I've never had to know that on a GMAT question). 
This definition does not imply that zero if a factor of itself. Please note the direction of your condition,  gmatophobia . "If a is a factor of b then..." so you're presupposing that zero is a factor of itself. If it isn't, then the conditional clause is moot.

ScottTargetTestPrep · Answer

There is no universal agreement on this, some sources say that 0 is a factor of 0 while other sources say that 0 is not a factor of any number, including 0. In my opinion, 0 should be considered as a factor of 0. If you try to apply the first definition with a = 0, notice that you don't get some non-integer quotient, you get nothing at all. Do not make the mistake of thinking that there is a number called "undefined" lurking somewhere on the number line. There is not, it means that that operation is not defined. So you can't even use the first definition to test whether 0 is a factor of 0. On the other hand, you can define a factor as "a is called a factor of b if there exists some integer k such that b = a * k". You can apply this definition with a = b = 0 and the conclusion will be that 0 is a factor of 0. However, as I mentioned earlier, there is no general consensus on this issue. I don't think GMAT will ever give you a question where you need to know whether 0 is a factor of 0 to answer it.

KarishmaB · Answer

gmatophobia

When we talk of factors, we usually mean positive integers. So the smallest factor of 10 is 1. 
0 and negative integers are not included in the definition of factors.

When we talk of multiples, we mean all integers. 0 is a multiple of every integer. 
So 0 is a multiple of 10. 
-10 is also a multiple of 10 but -10 is not a factor of 10 because factors are not negative.

Just keep this in mind and you should face no trouble in GMAT questions.

gmatophobia · Answer

Thank you avigutman ScottTargetTestPrep and KarishmaB for your valuable insights on this topic.

ScottTargetTestPrep · Answer

Of course!

Is Zero a factor of Itself

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