Lcm of 40, 144 and 180

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 40, 144, and 180. We will find the LCM by determining the prime factorization of each number and then using those factorizations to calculate the LCM.

Answer

The LCM of 40, 144, and 180 is $720$.

Steps to Solve

Find the prime factorization of each number

Start by determining the prime factorization of each value:

For 40: $$ 40 = 2^3 \times 5^1 $$
For 144: $$ 144 = 2^4 \times 3^2 $$
For 180: $$ 180 = 2^2 \times 3^2 \times 5^1 $$

Identify the highest power of each prime factor

List all prime factors from the three numbers and select the highest power for each:

For prime factor $2$: Maximum power from 40, 144, and 180 is $2^4$ (from 144).
For prime factor $3$: Maximum power from 144 and 180 is $3^2$ (from both).
For prime factor $5$: Maximum power from 40 and 180 is $5^1$ (from both).

Calculate the LCM using the highest prime factor powers

Now that we have the highest powers, we can find the LCM by multiplying these together:

$$ \text{LCM} = 2^4 \times 3^2 \times 5^1 $$

Perform the multiplication

Calculate the LCM step by step:

Calculate $2^4 = 16$.
Calculate $3^2 = 9$.
Now multiply: $$ 16 \times 9 = 144 $$
Finally, multiply by $5$: $$ 144 \times 5 = 720 $$

The least common multiple (LCM) of 40, 144, and 180 is $720$.

More Information

The least common multiple is the smallest non-zero number that is a multiple of all the given numbers. In this case, $720$ can be divided exactly by 40, 144, and 180.

Tips

Forgetting to consider the highest power of each prime factor in the LCM calculation.
Confusing LCM with greatest common divisor (GCD), which is focused on the common factors instead of the multiples.

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