What is the LCM of 36 64 72 96 and 120?
What is the LCM of 36, 64, 72, 96, and 120?
The LCM (Least Common Multiple) of a set of numbers is the smallest non-zero integer that is divisible by each of the numbers. To find the LCM of several numbers, you can use the prime factorization method or the division method. Here, we’ll use the prime factorization method. This involves breaking each number down into its prime factors, then taking the highest power of each prime that appears in these factorizations.
Step 1: Prime Factorization
Let’s start by finding the prime factors of each number:
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Prime factorization of 36:
- 36 = 2^2 \cdot 3^2
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Prime factorization of 64:
- 64 = 2^6
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Prime factorization of 72:
- 72 = 2^3 \cdot 3^2
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Prime factorization of 96:
- 96 = 2^5 \cdot 3^1
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Prime factorization of 120:
- 120 = 2^3 \cdot 3^1 \cdot 5^1
Step 2: Identify the Highest Powers of all Prime Factors
From the factorizations above, identify the highest power of each prime:
- For 2: The highest power is 2^6 (from 64).
- For 3: The highest power is 3^2 (from 36 and 72).
- For 5: The highest power is 5^1 (from 120).
Step 3: Calculate the LCM
Multiply these highest powers of each prime together to find the LCM:
\text{LCM} = 2^6 \cdot 3^2 \cdot 5^1
Now calculate the value:
2^6 = 64, \quad 3^2 = 9, \quad 5^1 = 5
Multiply these together:
\text{LCM} = 64 \cdot 9 \cdot 5
First, compute 64 \cdot 9:
64 \cdot 9 = 576
Next, multiply the result by 5:
576 \cdot 5 = 2880
Thus, the LCM of 36, 64, 72, 96, and 120 is 2880.
Verification
To verify, we check that 2880 is divisible by each of the numbers:
- 2880 \div 36 = 80, which is an integer.
- 2880 \div 64 = 45, which is an integer.
- 2880 \div 72 = 40, which is an integer.
- 2880 \div 96 = 30, which is an integer.
- 2880 \div 120 = 24, which is an integer.
Since 2880 is divisible by all the given numbers, our calculation is correct. The LCM of 36, 64, 72, 96, and 120 is indeed 2880.
Feel free to ask if you need help with anything else! @user