Factors and Multiples
Key Definitions
Factors are numbers that divide another number exactly (without a remainder).
Multiples are the products of a number multiplied by an integer.
Understanding Factors
Factors are numbers that divide evenly into another number. If A × B = C, then both A and B are factors of C.
Example: Finding Factors of 24
To find all factors of 24, we need to find all pairs of numbers that multiply to give 24:
1 × 24 = 24, so 1 and 24 are factors of 24
2 × 12 = 24, so 2 and 12 are factors of 24
3 × 8 = 24, so 3 and 8 are factors of 24
4 × 6 = 24, so 4 and 6 are factors of 24
Therefore, the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24
Tips for Finding Factors
- Systematic Approach: Start with 1 and go up to the square root of the number.
- Pairing: For each factor you find, its paired factor is the original number divided by it.
- Even Numbers: All even numbers have 2 as a factor.
- Divisibility Rules: Use divisibility rules to quickly identify potential factors.
Methods to Find Factors
Method 1: Systematic Division
Divide the number by integers starting from 1 up to the square root of the number. If division results in a whole number, both the divisor and the quotient are factors.
Method 2: Factor Tree
Create a factor tree to break down a number into its prime factors. All factors of the number can be formed by multiplying different combinations of these prime factors.
Example: Factor Tree for 36
36
/ \
2 18
/ \
2 9
/ \
3 3
Prime factorization: 36 = 2² × 3²
From this, we can find all factors by creating all possible combinations of these prime factors:
2⁰ × 3⁰ = 1
2¹ × 3⁰ = 2
2⁰ × 3¹ = 3
2² × 3⁰ = 4
2¹ × 3¹ = 6
2⁰ × 3² = 9
2² × 3¹ = 12
2¹ × 3² = 18
2² × 3² = 36
Therefore, the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36
Understanding Multiples
A multiple of a number is the product of that number and any integer. In other words, if A × B = C, then C is a multiple of both A and B.
Example: First 10 Multiples of 7
To find multiples of 7, we multiply 7 by consecutive integers:
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
7 × 5 = 35
7 × 6 = 42
7 × 7 = 49
7 × 8 = 56
7 × 9 = 63
7 × 10 = 70
Therefore, the first 10 multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, and 70
Properties of Multiples
- Every number has infinitely many multiples.
- All multiples of a number are divisible by that number.
- Zero is a multiple of every number (0 = n × 0 for any n).
- All multiples of 2 are even numbers.
- All multiples of 5 end in 0 or 5.
Common Factors and Multiples
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) or Greatest Common Divisor (GCD) of two or more numbers is the largest positive integer that divides each of the numbers without a remainder.
Example: Finding GCF of 48 and 60
Method 1: Listing Factors
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Common factors: 1, 2, 3, 4, 6, 12
GCF(48, 60) = 12
Method 2: Prime Factorization
48 = 2⁴ × 3¹
60 = 2² × 3¹ × 5¹
GCF = 2² × 3¹ = 4 × 3 = 12
Method 3: Euclidean Algorithm
60 = 48 × 1 + 12
48 = 12 × 4 + 0
GCF(48, 60) = 12
Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by each of the numbers without a remainder.
Example: Finding LCM of 12 and 18
Method 1: Listing Multiples
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, ...
Multiples of 18: 18, 36, 54, 72, 90, 108, ...
Common multiples: 36, 72, 108, ...
LCM(12, 18) = 36
Method 2: Prime Factorization
12 = 2² × 3¹
18 = 2¹ × 3²
LCM = 2² × 3² = 4 × 9 = 36
Method 3: Using GCF
LCM(a, b) = (a × b) ÷ GCF(a, b)
LCM(12, 18) = (12 × 18) ÷ GCF(12, 18) = 216 ÷ 6 = 36
Special Types of Numbers
Prime Numbers
A prime number is a number greater than 1 that has exactly two factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Composite Numbers
A composite number is a number greater than 1 that has more than two factors.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, ...
Perfect Numbers
A perfect number is a positive integer that is equal to the sum of its proper positive divisors.
Example: 6 is a perfect number because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6
Coprime Numbers
Two numbers are coprime (or relatively prime) if their GCF is 1.
Example: 15 and 28 are coprime because GCF(15, 28) = 1
Applications of Factors and Multiples
Reducing Fractions
To reduce a fraction to its simplest form, divide both the numerator and denominator by their GCF.
Example: Simplifying 24/36
GCF(24, 36) = 12
24 ÷ 12 = 2
36 ÷ 12 = 3
24/36 = 2/3
Finding Equivalent Fractions
To find equivalent fractions, multiply both the numerator and denominator by the same number.
Example: Finding Equivalent Fractions for 2/5
2/5 = (2×2)/(5×2) = 4/10
2/5 = (2×3)/(5×3) = 6/15
2/5 = (2×4)/(5×4) = 8/20
Adding and Subtracting Fractions
To add or subtract fractions with different denominators, convert them to equivalent fractions with the same denominator (the LCM of the original denominators).
Example: Adding 2/3 and 5/12
LCM(3, 12) = 12
2/3 = (2×4)/(3×4) = 8/12
2/3 + 5/12 = 8/12 + 5/12 = 13/12 = 1 1/12
Divisibility Rules
| Number | Divisibility Rule | Example |
|---|---|---|
| 2 | The last digit is even (0, 2, 4, 6, or 8) | 3846 is divisible by 2 because 6 is even |
| 3 | The sum of all digits is divisible by 3 | 423 is divisible by 3 because 4+2+3=9, and 9 is divisible by 3 |
| 4 | The last two digits form a number divisible by 4 | 5724 is divisible by 4 because 24 is divisible by 4 |
| 5 | The last digit is 0 or 5 | 2155 is divisible by 5 because it ends with 5 |
| 6 | The number is divisible by both 2 and 3 | 426 is divisible by 6 because it's divisible by both 2 and 3 |
| 7 | Multiply the last digit by 2, subtract from the rest of the number, and check if the result is divisible by 7 (repeat if needed) | 749: 74-(9×2)=74-18=56, 56 is divisible by 7, so 749 is divisible by 7 |
| 8 | The last three digits form a number divisible by 8 | 9624 is divisible by 8 because 624 is divisible by 8 |
| 9 | The sum of all digits is divisible by 9 | 8172 is divisible by 9 because 8+1+7+2=18, and 18 is divisible by 9 |
| 10 | The last digit is 0 | 5230 is divisible by 10 because it ends with 0 |
Interactive Tools
Factor Calculator
Enter a positive integer to find all its factors:
GCF and LCM Calculator
Enter two positive integers to find their GCF and LCM:
Factors and Multiples Quiz
Test Your Knowledge
Question 1: What are all the factors of 18?
Question 2: What is the LCM of 8 and 12?
Question 3: Which of the following is a prime number?
Question 4: What is the GCF of 36 and 60?
Question 5: Which of the following numbers is divisible by 9?