Least common multiple of 60 and 24

Understand the Problem

The question is asking to find the least common multiple (LCM) of the numbers 60 and 24. To solve this, we can use the prime factorization method or the division method to find the smallest multiple that both numbers share.

Answer

The LCM of 60 and 24 is $120$.

Steps to Solve

Find the prime factorization of 60

To factor 60, we divide it by the smallest prime number until we cannot divide anymore.

$60 \div 2 = 30$
$30 \div 2 = 15$
$15 \div 3 = 5$
$5 \div 5 = 1$

So, the prime factorization of 60 is: $$ 60 = 2^2 \times 3^1 \times 5^1 $$

Find the prime factorization of 24

Next, we will factor 24 in the same way.

$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
$3 \div 3 = 1$

Now, the prime factorization of 24 is: $$ 24 = 2^3 \times 3^1 $$

Determine the LCM using the highest powers of each prime

To find the LCM, we take the highest power of each prime factor from both factorizations:

From $2$: the highest power is $2^3$ (from 24)
From $3$: the highest power is $3^1$ (common to both)
From $5$: the highest power is $5^1$ (from 60)

Now we can multiply these together to get the LCM:

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

Calculate the LCM

Doing the multiplication:

$$ \text{LCM} = 8 \times 3 \times 5 $$

First, calculate $8 \times 3 = 24$. Then, multiply $24 \times 5 = 120$.

Thus, $$ \text{LCM} = 120 $$

The least common multiple (LCM) of 60 and 24 is $120$.

More Information

The least common multiple is the smallest number that is a multiple of both original numbers. In this case, 120 is not only a multiple of both 60 and 24, but also is derived from their prime factors.

Tips

Forgetting to include all prime factors at their highest powers can lead to incorrect LCM.
Confusing LCM with GCD (Greatest Common Divisor) is a common error. Remember, LCM is about finding multiples, while GCD is about common factors.

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