lcm of 12 and 32

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 12 and 32. To solve this, we will determine the multiples of each number and find the smallest value that appears in both sets.

Answer

The least common multiple of 12 and 32 is $96$.
Answer for screen readers

The least common multiple (LCM) of 12 and 32 is $96$.

Steps to Solve

  1. Find the Prime Factorization of Each Number

Start by finding the prime factors of 12 and 32.

For 12:

  • The prime factorization of 12 is: $$ 12 = 2^2 \times 3^1 $$

For 32:

  • The prime factorization of 32 is: $$ 32 = 2^5 $$
  1. Determine the Highest Powers of Each Prime Factor

Next, identify the highest power of each prime factor that appears in both factorizations.

  • For the factor 2: the highest power is $2^5$ (from 32).
  • For the factor 3: the highest power is $3^1$ (from 12).
  1. Multiply the Highest Powers Together

Now, multiply these highest powers together to find the LCM.

  • The LCM is calculated as: $$ \text{LCM} = 2^5 \times 3^1 $$
  1. Calculate the LCM

Now perform the multiplication:

$$ 2^5 = 32 $$ $$ 3^1 = 3 $$ $$ \text{LCM} = 32 \times 3 = 96 $$

The least common multiple (LCM) of 12 and 32 is $96$.

More Information

The least common multiple is the smallest number that is a multiple of both original numbers. In this case, 96 is the first number that both 12 and 32 divide evenly into. LCM is particularly useful in problems involving adding or subtracting fractions with different denominators.

Tips

  • A common mistake is to simply multiply the two numbers together (12 * 32) and consider that the LCM. This does not give the correct answer because the result may not be the smallest common multiple.
  • Another mistake is to forget to consider the highest powers of all prime factors involved.

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