LCM of 20 and 36

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 20 and 36, which is the smallest number that is a multiple of both 20 and 36.

Answer

$180$

Steps to Solve

Find the prime factorization of both numbers

First, we need to break down each number into its prime factors.

For 20:

The prime factors are $2^2 \times 5^1$

For 36:

The prime factors are $2^2 \times 3^2$

Identify the highest powers of each prime factor

Next, we find the highest powers of each prime that appear in the factorizations of both numbers.

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

Multiply the highest powers of each prime factor

Now we will calculate the LCM by multiplying the highest powers of each prime factor identified above.

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

Calculating this step-by-step:

First, calculate $2^2 = 4$.
Then, calculate $3^2 = 9$.
Finally, calculate $5^1 = 5$.

Now multiply them together:

$$ LCM = 4 \times 9 \times 5 $$

Perform the multiplication

Calculating it step-by-step:

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

So the final result is:

$$ LCM = 180 $$

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

More Information

The least common multiple is useful in many mathematical applications, including solving problems involving fractions, scheduling, and finding common denominators. The LCM helps ensure that calculations are performed on comparable terms.

Tips

Forgetting to consider all prime factors and their highest powers. To avoid this, always make sure to list out the prime factors clearly for each number.
Confusing the LCM with the greatest common divisor (GCD). Remember, the LCM is the smallest multiple, while the GCD is the largest shared factor.

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