Basics of Factors & Divisors

We already know that 15 is divisible by 3 because the result of 15 ÷ 3 is an integer. Similarly, 15 is divisible by several other integers. 15 is divisible…
by Talha Omer, M.Eng., Cornell Grad
in Math Concepts

We already know that 15 is divisible by 3 because the result of 15 ÷ 3 is an integer.

Similarly, 15 is divisible by several other integers.

15 is divisible by 1, 3, 5, and 15.

And 15 is also divisible by -1, -3, -5, and -15.

In total, 15 is divisible by 8 integers: 1, -1, 3, -3, 5, -5, 15, and -15.

All of these 8 integers (1, 3, 5, 15, -1. -3, -5, -15) are called factors of 15.

On the questions, just like the term divisible, the term factor is never used when there is a non-integer involved.

For example,

  • 1.5 ÷ 0.5 = 3 (0.5 is not a factor of 1.5, because 0.5 is not an integer)
  • 5 ÷ 2 = 2.5 (2 is not a factor of 5, because 5 is not divisible by 2)
  • -5 ÷ 2.5 = -2 (2.5 is not a factor of -5, because 2.5 is not an integer)

So simply put, a factor is an integer that fully divides another integer.

Is 100 a factor of 50?
  • Yes
  • No

Is 100 a factor of 50 is the same as saying: Is 50 ÷ 100 = an integer?

Since 50 ÷ 100 = 0.5, hence 100 is not a factor of 50.
Or we can also say that 50 is not divisible by 100.

 

Is 27 a factor of 0?

  • Yes
  • No

Is 27 a factor of 0 is the same as saying: Is 0 ÷ 27 = an integer?

Since 0 ÷ 27 = 0, hence 27 is a factor of 0.

Or we can also say that 0 is divisible by 27.

 

Is 0 a factor of 10?

  • Yes
  • No

Is 0 a factor of 10 is the same as saying: Is 10 ÷ 0 = an integer?

Since 10 ÷ 0 = undefined, hence 0 is not a factor of 10.

Or we can also say that 10 is not divisible by 0.

 

Which of the following is a factor of 0?

  • -5
  • -10
  • 0
  • 1
  • 2
  • 7

Every non-zero integer is a factor of zero (0).

We know that a factor of zero is an integer that fully divides zero.

All non-zero integers fully divide 0.

0 ÷ (-5) = 0
0 ÷ (-10) = 0
0 ÷ 1 = 0
0 ÷ 2 = 0
0 ÷ 7 = 0

However, 0 ÷ 0 = undefined. Therefore, every non-zero integer is a factor of zero.

 

On the questions, factor and divisor are exact synonyms.

Therefore, ±1, ±3, ±5, and ±15 are all divisors, as well as factors, of 15.

We can now come up with a series of statements, all of which mean the same thing:

  • 5 is a factor of 15
  • 5 is a divisor of 15
  • 15 is divisible by 5
  • 15 ÷ 5 = integer

The following generalized statements all mean the same thing:

  • x is a factor of y
  • x is a divisor of y
  • y is divisible by x
  • y/x = integer

where, of course, x must be a non-zero integer, y must be an integer, and the result of y ÷ x must be an integer.

Remember these examples – part of working integer problems is translating these confusing phrasings into simple pieces of information that can help us solve the problem.

In summary:

  • A factor is an integer that fully divides another integer.
  • The terms factor and divisor are the exact same thing.

SEE ALSO

GRE Math: Fundamentals of a Triangle

Like most people who take the GRE, your Math knowledge might be a little rusty. The good news is that GRE only tests you on math that you have already learned in high school. If you are worried that you have forgotten most of what you learned in high school, you will...

read more

How do you Calculate Standard Deviation on the GRE?

Standard deviation is an important statistical term tested on the GRE. It gives you an insight into the deviation or dispersion of a set of numbers from its mean; therefore, a low standard deviation implies that the numbers are very close to the mean, and vice versa....

read more

What are Prime Factors?

Every integer greater than 1 either is a prime number or a composite number. All composite numbers can be expressed as a product of prime numbers. For example, 6 can be expressed as 2 × 3. The prime factors of 6 are 2 and 3. Whereas the expression 2 × 3 is called...

read more

How to calculate Factors of Perfect Squares

Let's see how we use the chart to deal with this question: How many positive factors does 100 have? Set up the chart: 100 Left Column Right Column 1 100 2 50 4 25 5 20 10 10   Remember the rule - once the factors repeat themselves, (i.e. 10 and 10), stop. Since...

read more

What are Factor Tables?

Some integer questions may require you to find the Factors of an integer. The factor table is a basic technique for finding all the factors of any integer. This technique can also be useful for questions asking how many factors a particular integer has. The technique...

read more

Basics of Prime Numbers

Primes play a central part in integer questions. Overconfidence is dangerous here: while almost everybody can recite the definition of a prime number at the drop of a hat, the field is actually rife with misconceptions. We are here to make sure you know everything...

read more

Rules of Operations for Odd and Even Numbers

Let's formulate rules for addition, subtraction, multiplication, and division of Even and Odd numbers. Here are the rules for adding / subtracting even and odd numbers: Even ± Even = Even (e.g. 2 + 2 = 4; -4 - 2 = -6) Odd ± Odd = Even (e.g. 1 + 1 = 2; -31 - 1 = -32)...

read more

Basics of Even & Odd numbers

We'll begin with the basic definitions: Even: any integer that is divisible by 2. Examples: 2, 4, 14. Odd: any integer that is not divisible by 2, i.e. leaves a remainder of 1 when divided by 2. Examples: 1, 3, 5, 7, 9 So far so good. Consider this potential...

read more

Shortcuts to find Remainders

Find the remainder when 5,142,376,298 is divided by 9?     The long division method will definitely tell you the remainder of dividing 5,142,376,298 by 9 but you can very well imagine how long would that take. The good news is that just like the divisibility...

read more