GMAT Focus Edition Online Course: Number Properties
- Goalisb
- Jun 30
- 8 min read
Updated: Jul 11
In the world of numbers, relationships are everything. Just as addition and subtraction are inverse operations, so too are factors and multiples inextricably linked through the concept of divisibility.
A deep understanding of these two fundamental ideas, along with quick recognition of divisibility rules, is not merely helpful—it's absolutely critical for success on the GMAT. These concepts form the bedrock for prime factorization, Least Common Multiple (LCM), Greatest Common Divisor (GCD), and a myriad of number properties problems. This lesson will provide an exhaustive exploration, equipping you with the tools to confidently navigate any problem involving divisibility.
I. Factors: The Building Blocks of a Number
Imagine you have 12 cookies and you want to divide them equally among a group of friends, with no cookies left over. How many friends can be in the group? You could have 1 friend (they get 12), 2 friends (they get 6 each), 3 friends (they get 4 each), 4 friends (they get 3 each), 6 friends (they get 2 each), or 12 friends (they get 1 each). The numbers 1, 2, 3, 4, 6, and 12 are all factors of 12.
Formal Definition:
A factor (or divisor) of a positive integer N is any positive integer that divides N evenly (i.e., with a remainder of 0). When a is a factor of N, it means that N can be expressed as a * k for some other integer k. In this case, k is also a factor of N.
Key Characteristics of Factors:
Finite Count: Every positive integer (except 0, which has infinite factors) has a finite, countable number of factors.
Positive Factors: Unless specified, "factors" usually refer to positive factors. (Negative integers also have factors, e.g., -2 is a factor of 12 because 12 / -2 = -6). On the GMAT, focus on positive factors unless specified.
1 and N are Always Factors: The number 1 is a factor of every integer. Every integer is a factor of itself.
Factors are Less Than or Equal to N: All factors of a positive integer N are less than or equal to N.
How to Find All Positive Factors Systematically:
The most reliable way is to find factors in pairs, starting from 1.
Method:
Start with 1 and the number itself.
Test integers d starting from 2, checking if d divides N evenly.
If d divides N evenly, then d is a factor, and N/d is also a factor. List both as a pair.
Continue this process until d becomes greater than sqrt(N). Once d exceeds sqrt(N), you've found all unique pairs.
Example: Find all positive factors of 36.
1 * 36 = 36 (Factors: 1, 36)
2 * 18 = 36 (Factors: 2, 18)
3 * 12 = 36 (Factors: 3, 12)
4 * 9 = 36 (Factors: 4, 9)
5 does not divide 36 evenly.
6 * 6 = 36 (Factor: 6. Note: 6 is paired with itself).
Since 7 > sqrt(36)=6, we stop.
The positive factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36. (We will also learn to count factors using prime factorization in a later lesson).
II. Multiples: The Products of a Number
If you have a bus that carries 12 passengers, how many passengers can 1 bus carry? 2 buses? 3 buses? The answers are 12, 24, 36. These are all multiples of 12.
Formal Definition:
A multiple of an integer x is any integer that can be obtained by multiplying x by another integer. In simpler terms, M is a multiple of x if M is perfectly divisible by x (i.e., M / x is an integer with no remainder).
Key Characteristics of Multiples:
Infinite Count: Every non-zero integer has an infinite number of multiples.
Multiples are Greater Than or Equal to the Number: For positive integers, multiples of N are always greater than or equal to N.
Zero is a Multiple: 0 is a multiple of every non-zero integer because 0 * x = 0.
Negative Multiples: Multiples can also be negative (e.g., -12, -24 are multiples of 12 because 12 * (-1) = -12). On the GMAT, usually "multiples" refer to positive multiples unless specified.
How to Generate Multiples:
Simply multiply the given integer by 1, 2, 3, 4, ... (for positive multiples).
Example: List the first five positive multiples of 7.
7 * 1 = 7
7 * 2 = 14
7 * 3 = 21
7 * 4 = 28
7 * 5 = 35 The first five positive multiples of 7 are: 7, 14, 21, 28, 35.
III. The Interplay: Factors, Multiples, and Divisibility
These concepts are different sides of the same coin:
If A is a factor of B, then B is a multiple of A.
If A is a multiple of B, then B is a factor of A.
The phrase "A is divisible by B" means B is a factor of A, and A is a multiple of B.
Example:
5 is a factor of 20.
20 is a multiple of 5.
20 is divisible by 5.
IV. Divisibility Rules: Quick Checks for Factors
These rules allow for rapid determination of whether an integer is a factor of another without performing long division. Essential for efficiency on the GMAT.
Divisibility by 2: A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8 (i.e., it's an even number).
Example: 458 (Yes, ends in 8)
Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 741 (Sum = 7+4+1 = 12. 12 is divisible by 3, so 741 is divisible by 3).
Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Example:1 1,328 (28 is divisible by 4, so 1,328 is divisible by 4).
Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
Example: 9,070 (Yes, ends in 0)
Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Example: 2,148 (Ends in 8, so divisible by 2. Sum of digits = 2+1+4+8 = 15. 15 is divisible by 3, so 2,148 is divisible by 3. Since it's divisible by both 2 and 3, it's divisible by 6).
Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Example: 5,120 (120 is divisible by 8 (120 / 8 = 15), so 5,120 is divisible by 8).
Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 4,752 (Sum = 4+7+5+2 = 18. 18 is divisible by 9, so 4,752 is divisible by 9).
Divisibility by 10: A number is divisible by 10 if its last digit is 0.
Example: 780 (Yes, ends in 0)
V. Why Factors and Multiples are Crucial for the GMAT
Prime Factorization (Next Lesson): The entire concept of prime and composite numbers, and breaking numbers down into primes, relies on identifying factors.
Least Common Multiple (LCM) & Greatest Common Divisor (GCD): These are direct applications of factors and multiples and are frequently tested.
Fractions: Simplifying fractions involves finding common factors in the numerator and denominator. Finding common denominators (for addition/subtraction of fractions) involves multiples.
Word Problems: Many problems related to scheduling, quantities, sharing, or arrangements implicitly involve finding factors or multiples.
Number Properties: Questions about the number of factors, perfect squares, divisibility rules for complex expressions, and sequences often draw upon these concepts.
VI. Common Pitfalls to Avoid
Confusing Factors and Multiples: This is the most common mistake. Remember: Factors are SMALLER (or equal) to the number; Multiples are LARGER (or equal).
Forgetting 1 and the Number Itself: Always include 1 and the number itself when listing factors.
Forgetting Zero: 0 is a multiple of every non-zero integer.
Divisibility by 6: Remember to check for both 2 AND 3. If a number is divisible by 2 but not 3, it's not divisible by 6 (e.g., 8). If it's divisible by 3 but not 2, it's not divisible by 6 (e.g., 9).
Prime Numbers Only Have 2 Factors: By definition, 1 and themselves. This is crucial.
Interactive Check Your Understanding:
List all positive factors of 28.
List the first four positive multiples of 11.
Is 234 divisible by 6? Explain why.
If 5 is a factor of X, what does that imply about X in terms of multiples?
Practice Questions:
Which of the following numbers is a factor of 105?
a) 2
b) 3
c) 4
d) 6
How many positive factors does 40 have?
If a number is divisible by 4 and also by 9, what is the smallest number it must be divisible by?
A factory produces widgets in batches of 15. Which of the following could be the total number of widgets produced?
a) 100
b) 125
c) 150
d) 160
Which of the following is NOT a multiple of 12?
a) 0
b) 36
c) 72
d) 130
Solutions to Practice Questions:
Which is a factor of 105?
a) 2: No (105 is odd)
b) 3: Yes (1+0+5 = 6, divisible by 3)
c) 4: No (05 is not divisible by 4)
d) 6: No (Not divisible by 2) The correct answer is b) 3.
How many positive factors does 40 have?
List pairs:
1 * 40
2 * 20
4 * 10
5 * 8 (sqrt(40) is approx. 6.3, so stop at 5) The positive factors of 40 are {1, 2, 4, 5, 8, 10, 20, 40}. It has 8 positive factors.
Divisible by 4 and 9. Smallest number it must be divisible by?
If a number is divisible by both 4 and 9, it must be a multiple of both 4 and 9. Since 4 and 9 have no common factors other than 1 (they are "relatively prime"), the smallest number it must be divisible by is their product: 4 9 = *36**. (This is essentially finding the LCM, which we'll cover in the next lesson).
Total widgets produced (batch of 15)?
The total number of widgets must be a multiple of 15.
a) 100: Sum of digits is 1 (not div by 3)
b) 125: Not divisible by 3 (sum of digits 8)
c) 150: Divisible by 5 (ends in 0) and divisible by 3 (sum of digits 6). So, 150 / 15 = 10. Yes.
d) 160: Not divisible by 3. The correct answer is c) 150.
Which is NOT a multiple of 12?
a) 0: Yes (12 * 0 = 0)
b) 36: Yes (12 * 3 = 36)
c) 72: Yes (12 * 6 = 72)
d) 130: No. 130 / 12 is not an integer (12 X10 = 120, 12X 11 = 132). The correct answer is d) 130.