Integers: Factoring - Factors Vs. Multiples
One final note regarding zero:
Zero is never a factor of another integer, as dividing by zero is not defined.
Zero is a multiple of any integer. For example, zero is a multiple of 5 because you can write 0 as 5⋅0 - which is indeed 5 times an integer.
Thus, if x is a multiple of 5, then x could still equal 0 - unless the problem indicates that x cannot equal zero (e.g. x is positive).
It is sometimes easy to confuse factors and multiples, especially under the "fog of war" of the test, with questions whizzing past your head.
Factors and multiples are essentially opposite terms:
Factors of a number are integers that evenly divide into that number.
Multiples of a number are formed by multiplying that number by any integer.
Both factors and multiples can be positive or negative.Confused? Look at the following table, which contrasts the factors and multiples of 6:
| Factors of 6 are... |
Multiples of 6 are... |
|
| Examples: |
±1, ±2, ±3, and ±6 |
0, ±6, ±12, ±18, ±24....etc. |
| How to find: | Factors are those integers from 1 through 6 that divide evenly into 6. Then the negative versions of those integers as well. |
Multiply 6 by other Integers. |
| Greater/smaller |
factors ≤ 6 |
Can be smaller or larger |
Note that 6 is both a factor and a multiple of itself.
One last important bit, which again demonstrates the fact that factors and multiples are opposite terms:
If x is a factor of 6 then
6 is a multiple of x.
Example:
2 is a factor of 6, which means that 6 is a multiple of 2: 6=2⋅3.
Remember our confusing phrasings for Integers questions? We can add the final two phrasings involving multiples.
all of these GMAT question phrasings mean the same thing:
1) 5 is a factor of x
2) 5 is a divisor of x
3) x is divisible by 5
4) $$\frac{x}{5} = integer$$
5) $$x=5\times integer$$
6) x is a multiple of 5
All of the above basically tell you that x = 0, ±5, ±10, ±15, ±20, ±25...
