lcm 20 and 36
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 20 and 36. To find the LCM, we can list the multiples of each number or use prime factorization.
Answer
The least common multiple (LCM) of 20 and 36 is $180$.
Answer for screen readers
The least common multiple (LCM) of 20 and 36 is $180$.
Steps to Solve
- 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: $$36 = 2^2 \times 3^2$$
- 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)
- 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: $$ \text{LCM} = 4 \times 9 \times 5 $$
- 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.
The least common multiple (LCM) of 20 and 36 is $180$.
More Information
The least common multiple (LCM) is the smallest positive integer that is a multiple of both given numbers. It can be useful in solving problems involving fractions, where a common denominator is needed.
Tips
- Forgetting to include all prime factors when calculating the LCM.
- Using the lowest powers instead of the highest powers of the prime factors.
