Number Properties

To do well on the GMAT, you need to learn many different concepts related to the properties of numbers. Some will be easier to master than others, but we will cover them all here for you in the simplest way possible so that you can use logic to more quickly and accurately answer the math questions.

Number Line and Absolute Value

What is a number line? Plainly put, it is a line of numbers.

A number line will extend infinitely in two directions, one continuously toward infinity and the other continuously toward negative infinity. In other words, as you move to the right on the number line, values become larger. The farther you move to the left on the number line, the smaller the numbers become.

Zero separates the positive numbers from the negative numbers. For example, there is 3 and there is –3, there is 12 and there is –12, there is 1,385,388,905 and there is –1,385,388,905.

What is an absolute value?

The absolute value of any number is the number without its negative sign. What it really represents is the distance from zero on the number line. It is written simply as a number between two vertical lines. An example of this is:

|–5| = |+5| = 5

Look at it this way: –5 is 5 units from zero, so its absolute value is 5.

Properties of –1, 0, 1 and Other Numbers

What are the properties of zero? Can zero have properties or is it simply the lack of everything, a void, and therefore without properties? Well, the good news is that it can have properties and we will cover them here, along with the properties of some other numbers.

Adding or subtracting zero from any number does not change the value of the number.

0 + 7 = 7

9 + 0 = 9

8 – 0 = 8

Now, the rules change when you multiply by zero. In this case, the result of any number multiplied by zero is zero. See the following examples:

25 × 0 = 0

198 × 0 = 0

Dividing by zero cannot be done. If you put that sort of equation into your calculator, you will get an error.

Now that we have looked into the properties of 0, what about 1 and –1?

Multiplying or dividing any number by 1 does not change that number.

7 ÷ 1 = 7

9 × 1 = 9

–17 × 1 = –17

Things change slightly when we begin to multiply by –1. In this case, doing so changes the sign of the number are multiplying. Let's look at some examples.

z × (–1) = –z

9 × (–1) = –9

–6 ÷ (–1) = 6

To get the reciprocal of a number, we simply write 1 divided by that number. For example, the reciprocal of 6 is 1/6. With fractions, the reciprocal can also be found by interchanging the denominator and the numerator. Moreover, the reciprocal of a number between 0 and 1 is greater than the number itself. Here is an example:

As we can see, this figure is larger than the original number.

Conversely, the reciprocal of a number between 0 and -1 is less than the number:

For the square of a number, when it is a square of a number between 0 and 1, the reciprocal is much greater than that number.

Multiplying any negative number by a fraction between 0 and 1 will result in a number greater than the original number.

Operations and Signed Numbers

We looked at operations in a previous section, and now it is time to review them again. This time, we are looking at operations with regards to signed numbers (numbers with a + or - sign).

For addition of numbers with the same signs, we simply add them and keep the sign:

–9 + –2 = –11

However, if we are dealing with numbers with different signs, then we take the difference of the absolute values and keep the sign of the larger absolute value.

(–9) + (+3) = –6

Subtracting a number is the same as adding its inverse.

(–4) – (–8) = (–4) + (+8) = +4

In multiplication and division, the product or the quotient of the two numbers with the same sign is positive. Here is an example:

(–3) × (–4) = +12

–100 ÷ –50 = +2

The product or the quotient of two numbers with opposite signs is negative:

(-4) × (+3) = –12

–100 ÷ 50 = –2

Odd and Even Numbers

Odd and even numbers only apply to integers. Even numbers are integers divisible by 2, while odd numbers are not divisible by 2.

Any number that ends in 0, 2, 4, 6, or 8 are even, while integers ending in 1, 3, 5, 7, or 9 are odd. Negative numbers can also be odd or even. Zero is considered an even number.

When dealing with operations for odd and even numbers, there are some simple rules to follow:

• Odd + Odd = Even
• Even + Even = Even
• Odd + Even = Odd
• Odd × Odd = Odd
• Even × Even = Even
• Odd × Even = Even

Factors and Divisibility of Primes - The Facts

• Any integer divisible by another integer is effectively a multiple of that integer:

• 16 is a multiple of 4 because 16 ÷ 4 = 4 ; 4 × 4 = 16

• In division, when you are dealing with a remainder, the remainder is always smaller than the number we are dividing by.

• 17 divided by 3 is 5, and the remainder is 2

• A factor is a divisor of a number that can evenly divide into that integer.

• 9 has 3 factors: 1, 3, 9
• 1 × 9 = 9 ; 3 × 3 = 9

• The greatest common factor is the largest factor that can be shared by two numbers. For example, the common factors of 8 and 12 are 1, 2, and 4. The greatest common factor is therefore 4.

• One great thing about factors is that you can do divisibility tests to see if the number is divisible by 2, 3, 4, 5, 6, and 9. Here are the rules:

• Any number can be divided by 2 if the last digit is divisible by 2.
• 224 is divisible by 2 because 4 is divisible by 2.
• If the sum of all of the digits of a number is divisible by 3, then the entire number is divisible by 3.
• 135 is divisible by 3 because 1 + 3 + 5 = 9.
• A number is divisible by 4 if its last two digits are divisible by 4.
• 1,240 is divisible by 4 because 40 is divisible by 4.
• A number is divisible by 5 if its last digit is a 0 or a 5.
• 2,517,545 is divisible by 5.
• A number is divisible by 6 if it is divisible by BOTH 2 and 3.
• 4,422 is divisible by 6 because the last digit is divisible by 2, and the sum of its digits (4+4+2+2) is 12, which is divisible by 3.
• A number is divisible by 9 if the sum of the digits is divisible by 9, much like how the division rule for 3 works.
• 14,832 is divisible by 9 because sum of its digits (1+4+8+3+2) is 18, which is divisible by 9.

You can use these rules of divisibility to figure out factors and multiples of larger numbers as well. For example, you know that 12 does not go into 270 because the factor pairs of 12 are 6 ×2 and 4 × 3. If 270 is divisible by 6, 2, 4, and 3, then it is divisible by 12. That may seem like a lot of work and the long division may appear easier. But you don't have to test 2, because any number divisible by 4 is divisible by 2. You don't have to test 6 because 6 is 2 × 3. So just use the rules of 3 and 4. Since 70 is not evenly divisible by 4, 270 is not a multiple of 12.

Now let's move on to prime numbers. A prime number is any integer greater than 1 that can only be divided by 1 and itself. The first prime number is 2, and is the only even prime number. Here is a quick list of the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Prime factorization is the breaking down of a number into just its prime, rather than all, divisors. No matter how you begin the process of factorization, a number's prime factors will always be the same. For example, if you want the prime factor of 36, you can start with the prime number 2 and write:

So the prime factors of 36 are 2, 2, 3, and 3. If you start with the prime number 3, you still end up with the same result.

Determining the prime factorization for a number can be difficult, but there is an easy way. Let's look at this example:

150 = 10 × 15 = (2 × 5) × (5 × 3) = 2 × 5 × 5 × 3

First we break 150 into two numbers. There are multiple combinations, 10 × 15, 2 × 75, 3 × 50 ...etc. Any combination will do; let's pick 10 × 15

150 = 10 × 15

Next we factorize the two components.

10 × 15 = (2 × 5) × (5 × 3)

Great! We only have prime numbers in the equation. Therefore the prime factorization of 150 is

2 × 5 × 5 × 3.

However, if the question specified the distinct prime factors of 150, there are only three: 2, 3, and 5. Distinct means that you don't count duplicates.

Consecutive Numbers

If the numbers in a list are at a fixed interval, then they are consecutive numbers. They need to have a pattern to be considered consecutive numbers. Also, all the consecutive numbers that you encounter on the exam will be integers.

1, 2, 3, 4, 5… is consecutive in intervals of +1, adding up

2, 0, –2, –4, –6… is consecutive in intervals of –2, subtracting down

2, 4, 16, 256… is a consecutive in intervals of squaring

The concepts in this chapter can help you narrow down your answer choices and avoid careless mistakes. Before you start solving a question, pause and analyze the properties of the numbers in the question and the way each number relates to the others. You will find that sometimes you can quickly eliminate the odd or even numbers, the fractions, or the negative answer choices.

More Practice from the GMAT® Review 13th Edition: Questions 1, 7, 15, 28, 64 (Number properties)

More Practice from the GMAT® Review 13th Edition: Questions 59, 65, 78, 112, 132 (divisibility)