Factors

As promised in the last post, we will investigate the number of properties tested frequently on the GMAT. We first begin with prime numbers.

Prime Numbers

A number is prime if it has only two factors: ‘1’ and the number itself, e.g. ‘2’. A number with more than two factors is called composite, e.g., ‘4’. The concept of prime numbers applies only to positive integers. ‘1’ is itself neither prime nor composite. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31… Once you know these basic points about prime numbers, the next important concept is prime factorization.

Prime Factorization

Every positive integer can be written uniquely as a product of prime numbers, called its prime factorization.

How to do prime factorization

Let’s say we want to find the prime factors of 120. We start testing all integers to see if and how often they divide 120 and the subsequent resulting quotients evenly.

120 ÷ 2 = 60; save 2

60 ÷ 2 = 30; save 2

30 ÷ 2 = 15, save 2

15 ÷ 2 = 7.5, not evenly, so try the next highest number, 3

15 ÷ 3 = 5, save 3,

5 ÷ 5 = 1; save 5

Therefore 120 = 23 x 3 x 5

Using a “factor tree,” your work would look like this

Prime factorization helps us deduce a lot of things about the number. Let’s look at one of the most important deductions.

Number of factors

Once we have the prime factorization of a number, we can quickly find the total number of factors (this would otherwise be quite tedious since we want ‘all’ the factors).

Let’s say you are asked to find the total number of factors of 120.

First of all, the prime factorization of 120 = 23 x 3 x 5

To find the total number of factors

– Find the exponents of each prime number in the prime factorization ( here 3, 1, and 1)

– Add one to each value (so 4, 2, 2)

– Multiply the numbers

So the number of all factors of 120 = 4x2x2 = 16

Another variant is to find the number of odd and even factors. We leave out the exponent of 2 for odd factors since even a single ‘2’ on multiplication would give an even number. So, the number of odd factors = 2×2= 4. To find the number of even factors, we subtract the number of odd factors from the total number of factors (here, 16 – 4 = 12).

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