What is the lcm of 50 and 132

General
1

what is the lcm of 50 and 132

2

What is the LCM of 50 and 132?

Answer:

To find the least common multiple (LCM) of two numbers, we need to determine the smallest positive integer that is divisible by both numbers. There are methods to find the LCM, such as listing multiples, prime factorization, or using the greatest common divisor (GCD). I will use the prime factorization method, which is efficient and straightforward.

Step 1: Prime Factorization

Prime Factorization of 50:
Start by dividing 50 by the smallest prime number, which is 2:

  • 50 ÷ 2 = 25
    Next, divide 25 by the next smallest prime number, which is 5:
  • 25 ÷ 5 = 5
    Finally, 5 is a prime number, so we continue dividing by 5:
  • 5 ÷ 5 = 1
    Prime factors of 50 are: (2^1 \times 5^2)

Prime Factorization of 132:
Begin by dividing 132 by the smallest prime number, 2:

  • 132 ÷ 2 = 66
    Divide 66 by 2 again:
  • 66 ÷ 2 = 33
    Next, divide 33 by the next smallest prime number, 3:
  • 33 ÷ 3 = 11
    11 is a prime number:
  • 11 ÷ 11 = 1
    Prime factors of 132 are: (2^2 \times 3^1 \times 11^1)

Step 2: Determine the LCM

To find the LCM, take the highest power of each prime factor appearing in the factorizations.

  • For the prime factor 2:
    • Highest power = 2^2
  • For the prime factor 3:
    • Highest power = 3^1
  • For the prime factor 5:
    • Highest power = 5^2
  • For the prime factor 11:
    • Highest power = 11^1

Now, multiply these highest powers to find the LCM:

[
LCM = 2^2 \times 3^1 \times 5^2 \times 11^1
]

Calculate step-by-step:

  1. 2^2 = 4
  2. 5^2 = 25
  3. 4 \times 3 = 12
  4. 12 \times 25 = 300
  5. 300 \times 11 = 3300

Final Answer:

The least common multiple (LCM) of 50 and 132 is 3,300.