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GMAT Factors And Multiples

by ejkiv


July 17, 2015


Factors and multiples are ubiquitous on the GMAT, it seems that just about every question can either be solved or simplified by using these concepts.  While they are different concepts, I felt that they are so inter-related that they should be included together.

First, you must be able to distinguish between the two.  As far as the GMAT is concerned, factors are less than or equal to a number while multiples are greater than or equal to a number (This is for the GMAT only - we will discuss in the multiples section).

Factor: An integer (x) less than or equal to a number (y) that can be multiplied by another integer to get the number (y).  Or, x/y is an integer, so x is a factor of y.

In simpler terms the factors of 24 are numbers that divide evenly into 24 so that no remainder is produced.  These numbers are factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.  It is important to remember that 1 and the number itself are always factors of a number.  These can be easy to forget.

PRIME FACTORIZATION

This subset of factorization is of the utmost importance.  You will frequently need to find the prime factors of numbers.  In order to do this, you will need to build a ‘factor tree.‘   Two different factor trees for 36 are shown below:

Prime Factorization of 36

The underlined numbers are the prime factors of 36.  Essentially, what you have to do is  continue to find numbers that are divisible into 36, or the number you are factoring.  Once you get a prime number you no longer have to continue with that branch of the tree.  When all branches can no longer be factored, the numbers at the end of each branch are your prime factors.

The prime factors of 36 are then 2, 2, 3, 3.  However, do note that you may be asked to find distinct prime factors.  In this case, there are two distinct prime factors: 2 and 3.

You might write the prime factorization of 36 like this: (2^2)(3^2).

Divisibility Rules

 Divisibility Rules

Multiples

Traditionally, a multiple of any integer is the result when that integer is multiplied by another integer - this would include negative numbers and 0.  However, for GMAT purposes we are only interested in the positive numbers.

Multiple: An integer (x) that is the product of integer (y) and another positive integer.  In this case, x is a multiple of y. Or, x is a multiple of y if is x/y an integer.

For a tangible example, the multiples of 3 are all of the numbers that you get when multiplying 3 by a positive integer.  Here is a list:

3, 6, 9, 12, 15, 18, 21, 24, 27... and so on.

  Greatest Common Factor and Lowest Common Multiple

You may end up getting asked about the lowest common multiple (LCM) of two numbers.  There is a great way to figure this out by using venn diagrams.  Coincidentally, this method also helps to find the greatest common factor (GCF) of two numbers, as we mentioned earlier.  To use the venn diagram, simply put common values in the overlap and unique values outside the overlap.  If you were looking for the GCF and LCM of 18 and 24 you can use these steps:

    1. Find the prime factors of each number (18 - 2, 3, and 3.  24 - 2, 2, 2, and 3)
    2. Place the numbers appropriately in a Venn diagram.

Greatest Common Factor

3. Multiply the overlap for the GCF - in this case 2 x 3 = 6

4. Multiply all the numbers for the LCM - in this case 2 x 2 x 2 x 3 x 3 = 72

 This is a quick and easy way to calculate the GCF and LCM.  The LCM can also come in handy when adding/subtracting fractions.

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