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 smallest number that is a multiple of both 40 and 60.
Answer
$120$
Answer for screen readers
The least common multiple (LCM) of 40 and 60 is $120$.
Steps to Solve
- Prime Factorization of 40
To begin, we need to find the prime factors of 40. The prime factorization of 40 is: $$ 40 = 2^3 \times 5^1 $$
- Prime Factorization of 60
Next, we find the prime factors of 60. The prime factorization of 60 is: $$ 60 = 2^2 \times 3^1 \times 5^1 $$
- Identify the Highest Powers of Each Prime
Now, we take each prime factor and select the highest power from both factorizations.
- For prime 2: the highest power is $2^3$ (from 40)
- For prime 3: the highest power is $3^1$ (from 60)
- For prime 5: the highest power is $5^1$ (common in both)
- Calculate the LCM
Now, we multiply the highest powers of all prime factors to find the LCM: $$ \text{LCM} = 2^3 \times 3^1 \times 5^1 $$
Calculating this: $$ \text{LCM} = 8 \times 3 \times 5 = 120 $$
The least common multiple (LCM) of 40 and 60 is $120$.
More Information
The LCM is the smallest number that both original numbers can divide into without leaving a remainder. The LCM can also be useful in problems involving adding or subtracting fractions with different denominators.
Tips
- Incorrect Prime Factorization: Make sure to break down each number fully into its prime factors.
- Choosing the Wrong Powers: Sometimes the highest power can be overlooked; double-check each prime factorization.
