To find a number divisible by two numbers, you need to find a common multiple of those two numbers. The easiest way to do this is to find their least common multiple (LCM). Any multiple of the LCM will also be divisible by both original numbers.
Here's a breakdown of the process:
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Find the Least Common Multiple (LCM): The LCM is the smallest positive integer that is divisible by both numbers. There are a couple of ways to find the LCM:
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Listing Multiples: List the multiples of each number until you find a common multiple. For example, to find the LCM of 4 and 6:
- Multiples of 4: 4, 8, 12, 16, 20, 24,...
- Multiples of 6: 6, 12, 18, 24, 30,...
- The LCM of 4 and 6 is 12 (the smallest common multiple).
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Prime Factorization:
- Find the prime factorization of each number.
- Identify the highest power of each prime factor that appears in either factorization.
- Multiply these highest powers together to get the LCM.
Example: Find the LCM of 12 and 18.
- Prime factorization of 12: 22 x 3
- Prime factorization of 18: 2 x 32
- Highest power of 2: 22
- Highest power of 3: 32
- LCM(12, 18) = 22 x 32 = 4 x 9 = 36
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Finding Other Divisible Numbers: Any multiple of the LCM will also be divisible by the original two numbers. So, to find other numbers divisible by both, simply multiply the LCM by any positive integer.
- Example: Since the LCM of 4 and 6 is 12, the numbers 12, 24, 36, 48, 60, and so on, are all divisible by both 4 and 6.
Example:
Let's say you want to find a number divisible by both 3 and 5.
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Find the LCM of 3 and 5:
- Multiples of 3: 3, 6, 9, 12, 15, 18,...
- Multiples of 5: 5, 10, 15, 20,...
- The LCM of 3 and 5 is 15.
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Therefore, 15 is divisible by both 3 and 5.
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Other numbers divisible by both 3 and 5 include: 30 (15 x 2), 45 (15 x 3), 60 (15 x 4), and so on.
In essence, find the LCM, and then any multiple of that LCM will be divisible by the original numbers.
