Primes, LCM & GCD
IMAT Interactive Study Tool
Prime Numbers
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
Intuitive check: Can you divide the number evenly by anything other than 1 and itself? If no, it's prime.
- Examples: 2, 3, 5, 7, 11, 13, 17...
- The number 2 is the only even prime number.
- The number 1 is NOT a prime number.
Prime Factorization is the process of finding which prime numbers multiply together to make the original number. Every integer greater than 1 is either a prime number itself or can be represented as a product of prime numbers.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It's the biggest factor they have in common.
Method: Prime Factorization
To find the GCD of two numbers, find the prime factorization of each number. The GCD is the product of the lowest powers of all common prime factors.
Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of all of them. It's the first number that appears in all their "times tables".
Method: Prime Factorization
To find the LCM of two numbers, find the prime factorization of each number. The LCM is the product of the highest powers of all prime factors that appear in either factorization.
1. True or False: The number 1 is a prime number.
2. True or False: The LCM of two numbers is always greater than or equal to both numbers.
Example 1: Finding GCD and LCM
Find the GCD and LCM of 36 and 48.
Step 1: Find the prime factorization of each number.
\(36 = 2 \times 18 = 2 \times 2 \times 9 = 2^2 \times 3^2\)
\(48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2^4 \times 3^1\)
Step 2: Calculate the GCD.
The common prime factors are 2 and 3. We take the lowest power of each.
\(GCD(36, 48) = 2^2 \times 3^1 = 4 \times 3 = 12\)
Step 3: Calculate the LCM.
The prime factors are 2 and 3. We take the highest power of each.
\(LCM(36, 48) = 2^4 \times 3^2 = 16 \times 9 = 144\)
Conclusion: The GCD is 12 and the LCM is 144.
Example 2: LCM Word Problem
Two lighthouses flash their lights every 15 seconds and 18 seconds, respectively. If they flash together at 10:00 PM, at what time will they next flash together?
Step 1: Identify the concept.
We are looking for the first time an event will happen together again, which is a classic application of the Least Common Multiple (LCM).
Step 2: Find the prime factorization of each number.
\(15 = 3 \times 5\)
\(18 = 2 \times 9 = 2 \times 3^2\)
Step 3: Calculate the LCM.
Take the highest power of all prime factors (2, 3, and 5).
\(LCM(15, 18) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90\)
Step 4: Interpret the result.
The LCM is 90, which means they will flash together every 90 seconds (or 1 minute and 30 seconds).
Conclusion: They will next flash together at 10:01:30 PM.
| Concept | Definition | Prime Factorization Method |
|---|---|---|
| GCD | Largest common factor | Product of the lowest powers of common primes. |
| LCM | Smallest common multiple | Product of the highest powers of all primes. |