What is the least common multiple of 36 and 45?

Steps to Solve

1. Find the prime factorization of each number

First, we'll break down both 36 and 45 into their prime factors.

For 36:

• 36 can be divided by 2: $$ 36 = 2 \times 18 $$
• Then divide 18 by 2: $$ 18 = 2 \times 9 $$
• Finally, divide 9 by 3: $$ 9 = 3 \times 3 $$ Thus, the prime factorization of 36 is: $$ 36 = 2^2 \times 3^2 $$

For 45:

• 45 can be divided by 3: $$ 45 = 3 \times 15 $$
• Then divide 15 by 3: $$ 15 = 3 \times 5 $$ Thus, the prime factorization of 45 is: $$ 45 = 3^2 \times 5^1 $$

2. Determine the highest powers of each prime factor

Next, we'll take note of the highest powers of all prime factors from both factorizations.

• The prime factors we have are 2, 3, and 5.
• The highest power of 2 is $2^2$ (from 36).
• The highest power of 3 is $3^2$ (common to both).
• The highest power of 5 is $5^1$ (from 45).

3. Calculate the LCM using these prime factors

Now, we can calculate the LCM by multiplying the highest powers of each prime factor together: $$ \text{LCM} = 2^2 \times 3^2 \times 5^1 $$

Calculating that gives us: $$ \text{LCM} = 4 \times 9 \times 5 $$ First, calculate $4 \times 9 = 36$.

Then, multiply $36 \times 5 = 180$.

4. Conclusion with the LCM of the two numbers

Thus, the least common multiple (LCM) of 36 and 45 is: $$ \text{LCM} = 180 $$