Is -3 a multiple of 3?

This is part of a series on common misconceptions.

True or False?

\(-3\) is a multiple of \(3.\)

Why some people say it’s true: It’s \(-1\times 3.\)

Why some people say it’s false: Negative numbers aren't multiples, just as they aren't primes.

The statement is \( \color{green}{\textbf{true}}\).

Explanation: It is valid to multiply -1 by 3, get -3, and thus conclude -3 is a multiple of 3. Negative numbers can be multiples and it is valid to multiply them, so since -1 times 3 is -3, it follows that -3 is a multiple of 3. \(_\square\)

Rebuttal: This doesn’t work! You can’t multiply a negative. And even if you could, negative numbers can’t be primes nor composites, so they can’t be multiples.

Reply: You can multiply negatives and \(-1\times 3=-3\) is valid. Normal mathematical logic implies that -3 is a multiple of 3. While negative numbers are neither primes nor composites, this does not mean they can’t be multiples. For instance, 0 is neither a prime nor a composite, yet it has an infinite amount of factors. 0.5 is a multiple of 0.5, yet it is neither prime nor composite.

There is an incredibly big group of numbers like this, yet their having factors, while possible, is not always well-defined.