lcm of 40 and 32
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 40 and 32. To find the LCM, we can determine the prime factorization of both numbers and then use that information to calculate the LCM.
Answer
The least common multiple of 40 and 32 is $160$.
Answer for screen readers
The least common multiple (LCM) of 40 and 32 is $160$.
Steps to Solve
- Find the prime factorization of 40
To find the prime factors of 40, we can divide by the smallest prime number until we reach 1:
$$ 40 = 2 \times 20 $$
$$ 20 = 2 \times 10 $$
$$ 10 = 2 \times 5 $$
Thus, the prime factorization of 40 is:
$$ 40 = 2^3 \times 5^1 $$
- Find the prime factorization of 32
Similarly, we find the prime factors of 32:
$$ 32 = 2 \times 16 $$
$$ 16 = 2 \times 8 $$
$$ 8 = 2 \times 4 $$
$$ 4 = 2 \times 2 $$
So, the prime factorization of 32 is:
$$ 32 = 2^5 $$
- Determine the LCM using the prime factors
To find the LCM, take the highest powers of each prime factor from both numbers. The prime factors we have are 2 and 5.
For 2, the highest power is $2^5$ (from 32). For 5, the highest power is $5^1$ (from 40).
The LCM is given by:
$$ \text{LCM} = 2^5 \times 5^1 $$
Calculating that, we get:
$$ \text{LCM} = 32 \times 5 = 160 $$
The least common multiple (LCM) of 40 and 32 is $160$.
More Information
The least common multiple is useful in various mathematical applications, especially in finding common denominators for fractions. Knowing the LCM helps ensure that calculations involving multiple numbers stay consistent.
Tips
- Confusing GCD with LCM: Some confuse the greatest common divisor (GCD) with LCM. Remember, LCM is the smallest multiple that two or more numbers share, while GCD is the largest number that divides them both.
- Incorrect prime factorization: Incorrectly identifying the prime factors can lead to wrong calculations. Always double-check the factorization process.
