Prime numbers are integers that have exactly two divisors: “1” and “the number itself.”
Breaking an integer into a product of only prime numbers is called prime factorization.
What You Will Learn
Prime numbers continue in increasing order as follows: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
※ 1 is not included among the prime numbers.
When performing prime factorization, the key is to divide by small prime numbers in order, starting with 2, then 3, 5, 7, and so on.
Example Problem
Let us find the prime factorization of 120. Then, let us count how many divisors it has.
Solution:
① Divide by small prime numbers in order.
120 ÷ 2 = 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
② Write all the prime numbers used as a product.
120 = 2 × 2 × 2 × 3 × 5 ＝ 2³ × 3 × 5
③ To find the number of divisors, add 1 to each “number in the upper right” and multiply them together.
(3+1) × (1+1) × (1+1) = 4×2×2 = 16 divisors
Let us find the prime factorization of 120. Then, let us count how many divisors it has.
Solution:
① Divide by small prime numbers in order.
120 ÷ 2 = 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
② Write all the prime numbers used as a product.
120 = 2 × 2 × 2 × 3 × 5 ＝ 2³ × 3 × 5
③ To find the number of divisors, add 1 to each “number in the upper right” and multiply them together.
(3+1) × (1+1) × (1+1) = 4×2×2 = 16 divisors
Key Points
Remember
- For prime factorization, divide by small prime numbers (2, 3, 5, 7, …) in order.
- The number of divisors can be found by multiplying together “each number in the upper right + 1.”
- Using prime factorization, you can quickly find the greatest common divisor and the least common multiple.
Practice Problems
- Find the prime factorization of 84.
- Find the total number of divisors of 180.
- Find the greatest common divisor and the least common multiple of 2³×3²×5 and 2²×3×7.
Answers and Explanations
- Answer:84 = 2² × 3 × 7
Explanation: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, 7 ÷ 7 = 1. Therefore, 84 = 2² × 3 × 7. - Answer:18 divisors
Explanation: First, find the prime factorization: 180 = 4 × 45 = 2² × 3² × 5. Number of divisors = (2+1) × (2+1) × (1+1) = 3×3×2 = 18 divisors. - Answer:Greatest common divisor = 2² × 3 = 12, Least common multiple = 2³ × 3² × 5 × 7 = 2520
Explanation: For the greatest common divisor, take the smaller exponent of each prime factor: 2^min(3,2) × 3^min(2,1) × 5^min(1,0) × 7^min(0,1) = 2² × 3 = 12. For the least common multiple, take the larger exponent of each prime factor: 2³ × 3² × 5 × 7 = 8×9×5×7 = 2520.