lcm of 40 and 60

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 40 and 60. To solve this, we will find the multiples of each number and determine the smallest multiple that they share.

Answer

The least common multiple (LCM) of 40 and 60 is $120$.

Steps to Solve

Find the prime factorization of each number

First, we need to find the prime factors of 40 and 60.

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

For 60: $$ 60 = 2^2 \times 3^1 \times 5^1 $$

Identify the highest powers of each prime factor

Next, we identify the highest power of each prime factor from both factorizations.

For prime factor 2: the highest power is $2^3$ from 40.
For prime factor 3: the highest power is $3^1$ from 60.
For prime factor 5: the highest power is $5^1$ (appears in both).

So we collect these into one expression for the LCM.

Calculate the least common multiple

Now that we have the highest powers of all prime factors, we multiply these together to find the LCM:

$$ LCM(40, 60) = 2^3 \times 3^1 \times 5^1 $$

Calculating this gives:

$$ = 8 \times 3 \times 5 $$

Perform the multiplications step by step

Now we do the calculations step by step:

First, multiply 8 and 3:

$$ 8 \times 3 = 24 $$

Next, multiply the result by 5:

$$ 24 \times 5 = 120 $$

Thus, the least common multiple is 120.

The least common multiple (LCM) of 40 and 60 is $120$.

More Information

The least common multiple is useful in various applications including finding common denominators in fractions and analyzing periodic events. In this case, 120 is the smallest number that both 40 and 60 divide into without a remainder.

Tips

Failing to find the correct prime factorizations: Always double-check your prime factorization steps.
Forgetting to include all prime factors in the LCM calculation: It's essential to consider the highest power of all primes involved.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!