What is the least common multiple of 36 and 20?

Understand the Problem

The question is asking for the least common multiple (LCM) of the two numbers 36 and 20. To solve this, we will find the multiples of both numbers and identify the smallest common one.

Answer

The least common multiple (LCM) of 36 and 20 is $180$.

Steps to Solve

Find the prime factorization of each number

Start by breaking down both numbers into their prime factors.

For 36:

$36 = 6 \times 6 = 2 \times 3 \times 2 \times 3 = 2^2 \times 3^2$

For 20:

$20 = 4 \times 5 = 2 \times 2 \times 5 = 2^2 \times 5$

Identify the highest power of each prime factor

Now we need to take the highest powers of all the prime factors from both factorizations.

For prime factor $2$: Highest power is $2^2$ (from both 36 and 20).
For prime factor $3$: Highest power is $3^2$ (from 36).
For prime factor $5$: Highest power is $5^1$ (from 20).

Multiply these highest powers together to get the LCM

To find the least common multiple, multiply together the highest powers of all prime factors identified.

$$ LCM = 2^2 \times 3^2 \times 5^1 $$

Now, calculating this step-by-step:

First multiply $2^2 = 4$.
Then, $3^2 = 9$.
Finally multiply by $5$:

$$ 4 \times 9 \times 5 = 36 \times 5 = 180 $$

The least common multiple (LCM) of 36 and 20 is $180$.

More Information

The least common multiple is the smallest number that is a multiple of both given numbers. In this case, 180 is the first number that both 36 and 20 can divide evenly into. This concept is commonly used in solving problems related to fractions, schedules, and ratios.

Tips

Forgetting to use the highest power of each prime when calculating the LCM.
Confusing LCM with greatest common divisor (GCD), which is the largest number that divides both numbers.

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