# What is the exact meaning of "3 is not a factor of m or of n"?

3 is not a factor of m or of n.

Meaning of the above statement the way I understand it is 3 is not a factor of m or 3 is not a factor of n.

I think that I might be wrong because here I suggested an edit which was rejected.

My question:

Is the meaning of 3 is not a factor of m or of n the same as 3 is not a factor of both m and n. If yes, then please correct me by helping me understand what is it that I missunderstood.

Unfortunately, English does not always work in an entirely logical way, especially with 'logic words'. From a logical standpoint, "3 is not a factor of m or of n" should mean

3 is not a factor of m, OR 3 is not a factor of n. 3 might be a factor of m, and it might be a factor of n, but it is not a factor of both.

and "3 is not a factor of m and of n" should mean

3 is not a factor of m, AND 3 is not a factor of n. It can not be a factor of either.

This is actually backwards of the meaning in English. The English meaning of the first one is:

3 is not a factor of m, NOR is it a factor of n. It is not a factor of either.

and the second one means

3 is not a factor of both m and n. 3 might be a factor of m, and it might be a factor of n, but it is not a factor of both.

Think of it this way.

"Do you want ice cream or cake?" "No, I'm full." (I do not want ice cream, nor do I want cake.)

"Do you want ice cream and cake?" "No, I'll just take some cake." (I do not want ice cream and cake, but I do want some cake.)

That is the first reason your review was rejected. The second reason, is because "contraposition" is not a typo. According to wikipedia, contraposition is a law about contrapositives. Therefore, a contraposition argument is an argument using contrapositives and the law of contraposition, but a contrapositive argument doesn't really exist.

• Glad I could help! Apr 18, 2015 at 15:22
• This is plain wrong. From a logical standpoint, "three is not a factor of m or of n" parses exactly the same way as "I do not eat meat or fish." The negation applies to the disjunction, with the meaning "the following statement is not true: 'three is a factor of m, or three is a factor of n'." In logic and in English, this is equivalent to "three is not a factor of m and three is not a factor of n." Apr 19, 2015 at 0:22

Three is not a factor of m or of n = Three is not a factor of m, and three is not a factor of n. = Three is a factor of neither m nor n.

"Three is not a factor of both m and n" is not a good way to put this, because it could be taken to mean "three is not a common factor of m and n". Interpreted this way, it might still be possible for three to be a factor of either m or n individually if it is not a factor of the other one.

Here is my take on it.

``````    "3 is not a factor of m or of n"
=  clarify
"it is not true that 3 is a factor of m or of n"
=  clarify
"it is not true that 3 is a factor of m nor that 3 is a factor of n"
=  In mathematical notation
not ( m | 3 or n | 3)
=  De Morgan
not m | 3 and not n | 3
=  Back to English
"3 is not a factor of m and 3 is not a factor of n"
``````