lcm 20 and 36

Steps to Solve

1. Find the prime factorization of each number

Start by determining the prime factors of 20 and 36.

For 20:

• Divide by 2: $20 = 2 \times 10$
• Divide by 2 again: $10 = 2 \times 5$ Thus, the prime factorization of 20 is: $$20 = 2^2 \times 5^1$$

For 36:

• Divide by 2: $36 = 2 \times 18$
• Divide by 2 again: $18 = 2 \times 9$
• Divide by 3: $9 = 3 \times 3$ Thus, the prime factorization of 36 is: $$36 = 2^2 \times 3^2$$

2. Identify the highest powers of all prime factors

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

• For the prime factor 2: the highest power is $2^2$ (from both)
• For the prime factor 3: the highest power is $3^2$ (from 36)
• For the prime factor 5: the highest power is $5^1$ (from 20)

3. Calculate the LCM using the highest powers

Now, multiply the highest powers of all prime factors together: $$ \text{LCM} = 2^2 \times 3^2 \times 5^1 $$ Calculating this gives: $$ \text{LCM} = 4 \times 9 \times 5 $$

4. Simplify and find the final result

Now we will compute: $$ 4 \times 9 = 36 $$ Then, $$ 36 \times 5 = 180 $$

Thus, the least common multiple (LCM) of 20 and 36 is 180.