lcm of 42 and 36
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 42 and 36. To solve it, we can find the prime factors of both numbers, determine the maximum power of each prime factor that appears in their factorizations, and then multiply those together to get the LCM.
Answer
The least common multiple of 42 and 36 is $252$.
Answer for screen readers
The least common multiple (LCM) of 42 and 36 is 252.
Steps to Solve
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Find the prime factorization of 42 To factor 42 into its prime components, we divide by the smallest prime numbers: $$ 42 = 2 \times 3 \times 7 $$
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Find the prime factorization of 36 Similarly, we factor 36: $$ 36 = 2^2 \times 3^2 $$
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List the unique prime factors Identify all the unique prime factors from both factorizations:
- The unique prime factors are: 2, 3, and 7.
- Determine the maximum power of each prime factor Next, we find the highest power of each prime factor:
- For 2: The maximum power is $2^2$ from 36.
- For 3: The maximum power is $3^2$ from 36.
- For 7: The maximum power is $7^1$ from 42.
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Multiply the highest powers together Now we multiply these together to get the LCM: $$ \text{LCM} = 2^2 \times 3^2 \times 7^1 $$
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Calculate the value of LCM Finally, we calculate the value: $$ \text{LCM} = 4 \times 9 \times 7 = 252 $$
The least common multiple (LCM) of 42 and 36 is 252.
More Information
LCM is crucial in finding common denominators for fractions. Additionally, it's used in various applications like scheduling events that recur at different intervals.
Tips
- Forgetting to consider the highest power of each prime factor.
- Miscalculating when multiplying the prime factors together. Always double-check each multiplication step.
