What is the LCM of 32 and 40?

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 32 and 40. To solve this, we can use the prime factorization method or listing the multiples of each number to find the smallest multiple that they share.

Answer

The least common multiple of 32 and 40 is $160$.

Steps to Solve

Find the Prime Factorization of Each Number

Start by determining the prime factors of 32 and 40.

For 32:

Divide by 2: $32 \div 2 = 16$
Divide by 2: $16 \div 2 = 8$
Divide by 2: $8 \div 2 = 4$
Divide by 2: $4 \div 2 = 2$
Divide by 2: $2 \div 2 = 1$

Thus, the prime factorization of 32 is: $$ 32 = 2^5 $$

For 40:

Divide by 2: $40 \div 2 = 20$
Divide by 2: $20 \div 2 = 10$
Divide by 2: $10 \div 2 = 5$
Divide by 5: $5 \div 5 = 1$

Thus, the prime factorization of 40 is: $$ 40 = 2^3 \times 5^1 $$

Identify the Highest Powers of Each Prime Factor

Next, identify the highest power of each prime factor present in both factorizations.

From $32 = 2^5$ and $40 = 2^3 \times 5^1$, we get:

For the prime number 2, the highest power is $2^5$.
For the prime number 5, the highest power is $5^1$.

Calculate the Least Common Multiple (LCM)

To find the LCM, multiply the highest powers of all prime factors together.

So, we have: $$ LCM(32, 40) = 2^5 \times 5^1 $$

Calculating this gives: $$ LCM(32, 40) = 32 \times 5 = 160 $$

The least common multiple (LCM) of 32 and 40 is $160$.

More Information

The least common multiple is often used in solving problems that involve adding or subtracting fractions with different denominators. Understanding how to find the LCM can be very useful in various mathematical applications.

Tips

Confusing LCM with GCD (greatest common divisor). Always ensure you are looking for the least common multiple, not the greatest.
Misinterpreting the prime factorization process. It's important to accurately factor each number completely.

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