Factors: The Building Blocks of Numbers
What Exactly is a Factor?
In simple terms, a factor is a whole number that you can multiply by another whole number to get a specific product. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
Each of these multiplications results in the number 12, and when you divide 12 by any of these factors, the remainder is zero. This relationship is fundamental to number theory.
Methods for Finding All Factors
Finding all factors of a number can be done systematically. The most reliable method is the factor pair method.
Step 1: Start with 1 and the number itself. These are always factors.
Step 2: Test for divisibility by 2, 3, 4, etc. Check if the number can be divided evenly. If yes, both the divisor and the quotient are factors.
Step 3: Stop when the quotient is equal to or less than the divisor. This prevents repetition.
Let's find all factors of 24:
- 24 ÷ 1 = 24 → Factors: 1 and 24
- 24 ÷ 2 = 12 → Factors: 2 and 12
- 24 ÷ 3 = 8 → Factors: 3 and 8
- 24 ÷ 4 = 6 → Factors: 4 and 6
- 24 ÷ 5 is not a whole number, so 5 is not a factor.
- We stop here because the next divisor, 6, is already in our list.
So, the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
Prime Numbers and Prime Factorization
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and 13. Every whole number greater than 1 is either prime or can be written as a unique product of prime numbers. This is called its prime factorization.
The most common method for prime factorization is the factor tree. Let's find the prime factorization of 60.
We start by breaking 60 into two factors. We can use 6 × 10.
Then, we break down the composite factors: 6 = 2 × 3 and 10 = 2 × 5.
All the factors at the bottom are prime numbers. So, the prime factorization of 60 is $2^2 \times 3 \times 5$.
This is often written using exponents to group the same prime factors together.
| Number | Prime Factorization | With Exponents |
|---|---|---|
| 24 | 2 × 2 × 2 × 3 | $2^3 \times 3$ |
| 50 | 2 × 5 × 5 | $2 \times 5^2$ |
| 81 | 3 × 3 × 3 × 3 | $3^4$ |
| 100 | 2 × 2 × 5 × 5 | $2^2 \times 5^2$ |
Greatest Common Factor (GCF) and Least Common Multiple (LCM)
Factors are essential for finding the Greatest Common Factor (GCF)[1] and the Least Common Multiple (LCM)[2] of two or more numbers.
Greatest Common Factor (GCF): The largest factor that two or more numbers share.
Example: Find the GCF of 18 and 24.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6. The greatest is 6. So, GCF(18, 24) = 6.
Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers. While this is about multiples, we use prime factors to find it efficiently.
Example: Find the LCM of 12 and 18.
Prime factors of 12: $2^2 \times 3$
Prime factors of 18: $2 \times 3^2$
To find the LCM, take the highest power of each prime: $2^2 \times 3^2 = 4 \times 9 = 36$. So, LCM(12, 18) = 36.
Factors in Action: Real-World Applications
Factors are not just abstract math concepts; they are used constantly in daily life and various professions.
1. Organizing Items: Imagine you have 36 cookies and want to pack them into boxes so that each box has the same number of cookies and no cookies are left over. The possible arrangements depend on the factors of 36.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
You could have 1 box of 36, 2 boxes of 18, 3 boxes of 12, and so on.
2. Simplifying Fractions: To simplify a fraction like $\frac{24}{36}$, you find the GCF of the numerator and denominator. We already found that the GCF of 24 and 36 is 12. Dividing both by 12 gives us $\frac{24 \div 12}{36 \div 12} = \frac{2}{3}$.
3. Tiling a Floor: If you need to tile a rectangular floor that is 60 inches by 84 inches with square tiles, what is the largest size of square tile you can use without cutting? You need to find a number that divides evenly into both 60 and 84—this is the GCF. The GCF of 60 and 84 is 12, so the largest square tiles would be 12 inches by 12 inches.
Common Mistakes and Important Questions
Q: Is the number 1 a prime number?
No. A prime number must have exactly two distinct factors: 1 and itself. The number 1 has only one factor (itself), so it is not considered a prime number. It is a unique number called a unit.
Q: Can a number be a factor of itself?
Yes. Every number is a factor of itself because any number divided by itself equals 1, which is a whole number, with a remainder of 0. For example, 15 is a factor of 15 since 15 ÷ 15 = 1.
Q: What is the difference between a factor and a multiple?
Factors are numbers we multiply to get another number. Multiples are what we get after multiplying the number by an integer. For example, factors of 10 are 1, 2, 5, 10. Multiples of 10 are 10, 20, 30, 40, etc. A number has a finite number of factors but an infinite number of multiples.
Footnote
[1] GCF (Greatest Common Factor): The largest whole number that is a factor of two or more given numbers. Also known as the Greatest Common Divisor (GCD).
[2] LCM (Least Common Multiple): The smallest whole number that is a multiple of two or more given numbers.