What is the lowest common multiple?

Question

what is the lowest common multiple of 120 and 150? Ive got 240 but I'm pretty sure this is incorrect. could you show me how you work it out please?

Answer 1

I'll do it for $25$ and $40$.

1. Write out the prime factorization of $25$: $25 = 5^2$.
2. Write out the prime factorization of $40$: $40 = 8 \cdot 5 = 2^3 \cdot 5$.
3. To find the LCM, take the larger/largest exponent on each prime: $2^3 \cdot 5^2 = 200.$

Or using the formula user236182 suggested:

1. The GCF takes the smaller/smallest exponent on each prime instead of the largest, so $GCF(25,40) = 2^0 \cdot 5^1 = 5$.
2. Then, $LCM(25,40) = 25 \cdot 40 / 5 = 1000 / 5 = 200$.

So, there are two ways.

Can you do it for yours?

Answer 2

It might help your understanding to think about the tables of multiplication, which you might have learned up to 10 or 12. For example, $2 \times 1 = 2$, $2 \times 2 = 4$, $2 \times 3 = 6$, etc.

In the table of multiplication for 120, we have $120 \times 1 = 120$, $120 \times 2 = 240$, $120 \times 3 = 360$, etc., while for 150 we have $150 \times 1 = 150$, $150 \times 2 = 300$, $150 \times 3 = 450$, etc.

So we do have 240 in the table of multiplication for 120, but it does not occur in the table of multiplication for 150. The least common multiple is the smallest positive multiple that occurs in the tables of multiplication for both numbers. That would be $120 \times 5 = 150 \times 4 = 600$.

Of course it's inefficient to find the LCM by calculating the tables of multiplication for each number in each pair you're given and then looking for the smallest number in both tables. If you know how to calculate the greatest common divisor (GCD), then you can use it to calculate the LCM: $$\textrm{lcm}(a, b) = \frac{ab}{\gcd(a, b)}.$$ Thus in your example we have $$\textrm{lcm}(120, 150) = \frac{18000}{30} = 600.$$

You can also do it by prime factorization, but if you know the Euclidean algorithm for the GCD, you can use the formula above and then you don't need to factorize either number.