Number Properties - GMAT Math Study Guide
- Integers - The set of numbers including: ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4...
The following numbers are examples of numbers that are not integers: -1.56, 2/3, 105.1
- Units Digit - The number immediately to the left of the decimal point.
For example, the units digit of 652 is 2 and the units digit of 31 is 1.
- Factor (aka Divisor) - A factor of x is a number that evenly divides into x.
For example, 5 is a factor of 10 because 10/5 = 2. 15 is a factor of 45 because 45/15 = 3. 7 is not a factor of 10 because 10/7 ≠ integer.
- Multiple - A multiple of x is a number that is the product of x and an integer.
For example, the multiples of 3 are 3, 6, 9, 12, ... while the multiples of 5 are 5, 10, 15, 20, ...
- Prime Number - A number whose only factors are 1 and itself. Said differently, a number whose only integer divisors are 1 and itself.
For example, the number 3 is a prime number because its only factors are 1 and itself while the number 9 is not prime since its factors are 9, 1, and 3.
- Parity - Whether a number is even or odd.
For example, the parity of 2 is even while the parity of 3 is odd.
- Greatest Common Factor (GCF) (aka Greatest Common Divisor, GCD) - The GCF of x and y is the largest number that divides without a remainder into x and y.
For example, the GCF of 12 and 18 is 6 since 6 is the largest factor of both 12 and 18.
- Least Common Multiple (LCM) - The smallest multiple of two numbers that is perfectly divisible by each of the two numbers.
For example, the LCM of 6 and 9 is 18 since 18 is the smallest multiple of both 9 and 6 that is divisible by both 9 and 6.
Divisibility / Factors / Multiples
In order to perform division, if x (the dividend) is divided by y (the divisor), determine the number of times y will go into x. If x can be divided evenly by y, x is said to be divisible by y. (As will be noted later, y is also said to be a factor of x). However, if y is not a divisor (or factor) of x, then there will be a remainder left over. If the solution is not an integer (i.e., the solution has a remainder) or if x and y are large numbers, long division should be used. (For more information, see the division study guide).
At times, it is not important to know the exact quotient from a division operation (i.e., the exact number of times y will go into x). Instead, some mathematics concepts and problems simply require determining whether y will go into x (i.e., whether x is divisible by y). Fortunately, a number of shortcut rules exist for determining this and it is not always necessary to perform long-hand calculations. The following rules can save a tremendous amount of time.
Basic rules about divisors:
2 will be a divisor (or factor) of x if x is even
3 will be a divisor (or factor) of x if the sum of x's digits is divisible by 3
4 will be a divisor (or factor) of x if x can be divided by 2 twice (still even after divided by 2 once)
5 will be a divisor (or factor) of x if x's last digit is 0 or 5
6 will be a divisor (or factor) of x if x is divisible by both 2 and 3
8 will be a divisor (or factor) of x if x can be divided by 2 thrice (still even after dividing by 2 twice)
9 will be a divisor (or factor) of x if the sum of x's digits is divisible by 9
10 will be a divisor(or factor) of x if x's last digit is 0
Note: These divisibility rules are extremely important to memorize for the test. Due to time constraints on the test, questions involving a factor or prime number often require you to have memorized these rules.
Factors & Multiples
A factor of x must divide evenly into x. A multiple of x is any number that can be created by multiplying x by an integer.
Factors must divide into the integer and must be less than or equal to the integer. As a result, there are a finite number of factors. Multiples are results of the integer multiplied by a whole number and must be equal to or larger than the integer. Consequently, there are an infinite number of multiples. For example:
Factors: 1, 2, 5, 10 are all of the factors of 10 (i.e., the set consisting of the factors of 10 is finite).
Note: A number is a factor of itself
Multiples: 10, 20, 30, 40, 50, and 60 are a portion of the infinite possible multiples of 10.
If y does not evenly divide into x, then there will be a remainder left over. The remainder is the difference between the largest multiple of y which is lower than x, and x. For example:
3 does not go evenly into 29. Thus, a remainder will be left over. The closest multiple of 3 which is less than 29 is 27 (=3*9). 29-27 = 2. Therefore, 29/3 = 9 + remainder of 2 = 9 + (2/3).
Factors and Divisibility
An important consequence of the aforementioned properties is that if two numbers are both divisible by x, the difference, sum, and product will also be divisible by x. For example:
10 and 25 are both divisible by 5. Therefore:
10 - 25 = -15 is divisible by 5
10 + 25 = 35 is divisible by 5
10 * 25 = 250 is divisible by 5
Products & Factors
The product of x and y is divisible by each factor of x and each factor of y.
For example, this property means that the product of 4 and 18 (i.e., 72) is divisible by all the factors of 4 and all the factors of 18.
A prime number is an integer greater than 1 whose only factors are 1 and itself. Said differently, a prime number is one that is only divisible by 1 and itself.
Due to the prevalence of prime numbers on more difficult mathematics questions, it is helpful to memorize the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
5 is a prime number as its only factors are 1 and 5
9 is not a prime as it has factors 1, 3, and 9
Two important points should be reiterated about prime numbers as they show up quite frequently:
- The number 1 is not a prime number.
- The number 2 is the only even prime number.
The prime factorization of a number is the expression of that number as the product of its prime factors. In order to find the prime factor of a number, break down that number's factors until only prime numbers are left.
In order to determine a number's prime factorization, it is best to draw a prime factor tree. This diagram recursively breaks down a number into its factors. In other words, a prime factor tree takes any pair of factors for the number in question and splits these factors until all the factors are prime. For example:
What is the prime factorization of 40?
- Draw a prime factor tree
Underneath each number are two numbers that are factors whose product is the above number. For example, 2 and 20 lie beneath 40 since 2*20 = 40. Similarly, 5 and 2 lie beneath 10 since 5*2 = 10
- At the bottom level of each node (or leaf), the number is prime. This is the sign that the factor tree is complete. The prime factorization of 40 is thus: 5*2*2*2
As a result of the properties of factors, it makes no difference which factors you choose to begin your prime factor tree. There are only two things that matter in constructing a prime factorization tree:
- The product of the two factors beneath a number is that number (i.e., that number above the factors).
- The process of breaking down factors continues until prime numbers lie at the bottom of each node.
What is the prime factorization of 40?
Notice that, once again, the bottom of the prime factorization tree is composed of three 2s and one 5.
Factors and Primes
One consequence of the properties of prime factorization is that all the factors of a number can be constructed by using the prime factorization. This is best illustrated through an example:
List all the factors of 60.
- Find the prime factorization of 60.
Prime Factorization of 60: 2*2*3*5
- By arranging the factors of 60 into every possible combination, one can find all the factors of 60.
2*2 = 4
2*2*3 = 12
2*2*3*5 = 60
2*2*5 = 20
2*3 = 6
2*5 = 10
2*3*5 = 30
Parity: Odds and Evens
A number is even if it is divisible by 2 and odd if it is not divisible by 2. For example:
Evens: 2, 4, 6, 8, 10, ...
Odds: 1, 3, 5, 7, 9, ...
Odds and evens follow special rules under addition/subtraction, multiplication, and division.
Addition and Subtraction:
Adding two evens or adding two odds results in an even number. 2 + 6 = 8; 3+5 = 8
Adding an even and an odd results in an odd number. 1 + 2 = 3
If the sum of two prime numbers is odd, one of the prime numbers must be 2. This is because in order for a sum to be odd, one of the numbers must be even and the only even prime number is 2.
An odd number times an even or an even times an even results in an even number. 2 * 3 = 6; 2 * 2 = 4
An odd times an odd results in an odd number 3 * 5 = 15
An important implication of this is that any number with a prime factorization that includes 2 must be even. Similarly, any odd number will only have odd prime factors.
There are no general rules as the division of two integers may result in a fraction.
10/5 = 2
2/10 = 1/5
Greatest Common Factor and Least Common Multiple
Greatest Common Factor
The greatest common factor (GCF) is the largest number that divides two numbers evenly. To find the GCF:
- Find the prime factorization of each number.
- Multiply all of the common (or shared) prime factors together, using the lower power of repeated factors.
140 prime factors: 2, 2, 5, 7
42 prime factors: 2, 3, 7
common factors: 2, 7
GCF = 2 * 7 = 14 (Note: Do not include 2 twice; Use the lower power of repeated factors).
Least Common Multiple
The least common multiple (LCM) is the smallest number that is a multiple of two numbers. To find the LCM:
- Find the prime factorization of each number.
- Find the product of all of the prime factors. For prime factors in each number, do not use (i.e., multiply) the factor twice but instead use the greater power of the factor.
140 prime factors: 2, 2, 5, 7
42 prime factors: 2, 3, 7
LCM: 2 * 2 * 3 * 5 * 7 = 420 (Note: 2 appears twice--not thrice--because you use the highest power of a shared factor not all the instances of that factor).
Sign: Positive and Negative Numbers
Similar to odd and even numbers, positive(+) and negative(-) numbers have specific rules about the results during multiplication and division.
(-) * (+) = - or (+) * (-) = -
(-) * (-) = +
(+) * (+) = +
(-) ÷ (+) = - or (+) ÷ (-) = -
(-) ÷ (-) = +
(+) ÷ (+) = +
Note that zero is neither positive nor negative.
The units digit of a number is the number to the right of the tens place. For example, 6 is the units digit of 76 and 4 is the units digit of 894. As a result of the way multiplication and division occur, the units digit has interesting properties in multiplication and division.
Units Digit and Multiplication
The units digit of a product is the units digit of the product of the units digits of the two numbers being multiplied. This is likely too abstract and confusing, so consider the following example:
What is the units digit of the product of 7056*99463?
In a timed test environment, there is not enough time to perform this calculation by hand. However, there is a shortcut. The units digit of the product will be the units digit of 6*3, which is 8 since 6*3 = 18.
Counting Members of a Set
In order to count the members of a set of consecutive integers, one must:
- Subtract the largest number from the smallest number.
- Add one since the question asks not for the difference between the largest and smallest members of the set, but the number of integers belonging to the set.
Set Y consists of the consecutive integers from 0 to 1. How many members are there of set Y?
Due to the small size of the set, the answer is 2 without performing any calculations (i.e., integers 0 and 1).
It would be wrong to answer: 1 - 0 = 1
Instead, the right answer is: 1 - 0 + 1 = 2
Consider another example:
Set X consists of the consecutive integers from 134 to 246. How many members are there of set X?
It would be wrong to answer: 246 - 134 = 112
Instead, the right answer is: 246 - 134 + 1 = 113
Factors & Divisibility of Consecutive Integers
A list of consecutive integers of length x must have a multiple of x among its members. As a result, x is a factor of one of the members of the list.
The above formulation is likely confusing due to its abstract nature. The bottom line is that if you have a list of 5 consecutive integers (or any other length, x), 5 must be a factor of at least one of the members of that list of 5 numbers. If you doubt this property of consecutive integers, attempt to construct a list of 5 integers that does not contain at least one number for which 5 is a factor.
3, 4, 5, 6, 7 · 5 is a factor of 5
12, 13, 14, 15, 16 · 5 is a factor of 15
Notice that the property holds true if x = 3 instead of 5
4, 5, 6 · 3 is a factor of 6
8, 9, 10 · 3 is a factor of 9
An Important Implication
An important implication of the above number property is:
the product of a list of x consecutive integers is divisible by x
Consider the following example:
If x is a positive integer, the series x, x+1, x+2 must be divisible by which numbers?
Since the list consists of 3 integers, the list must contain a number that has a factor of 3. Consequently, the product of the integers in the list must contain a factor of 3 and therefore the product must also be divisible by 3.
Since the list also contains 1 and 2 integers, the product of the list's members will also be divisible by 1 and 2.
Types of GMAT Problems
- Factor Problems
Many times GMAT questions will ask how many times x appears in the prime factorization of y. The easiest method is to determine the prime factorization of y. Then, count how many times x appears.
In the prime factorization of 132, how many 2s are there?
Correct Answer: C
- Break 132 down into its prime factorization.
132 = 2*66 = 2 * (33 * 2) = 2 * ([3 * 11] * 2) = 2 * 2 * 3 * 11
- There are 2 twos.
- Odd/Even Combined with Positive/Negative Rules
When combining the rules for odd/even numbers with the rules for positive/negative numbers, it is important to also keep in mind the order of operations, PEMDAS.
Determine the correct characteristics of the following number:
Correct Answer: E
- Determine the sign and parity of the numerator and denominator.
Numerator: (+odd) * (-even) = -even
Denominator: (+odd) + (+odd) = +even
- However, there is a problem. There is no universal rule for determining the parity of a division operation. Consider the following:
-2/2 = -1 = odd
-4/2 = -2 = even
-10/4 = -2.5 = neither even nor odd
- The parity of the division equation cannot be determined.