I am thinking of a number. Three factors of my number are 15, 12, and 10. What is the smallest number I could be thinking of?
Understand the Problem
The question is asking for the smallest number that has 15, 12, and 10 as factors. To solve it, we need to find the least common multiple (LCM) of these three numbers.
Answer
The smallest number is 60.
Answer for screen readers
The smallest number that has 15, 12, and 10 as factors is 60.
Steps to Solve
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Identify the factors
The factors given are 15, 12, and 10. We need to find the least common multiple (LCM) of these numbers. -
Prime Factorization
Next, we perform the prime factorization of each number:- For 15: ( 15 = 3^1 \times 5^1 )
- For 12: ( 12 = 2^2 \times 3^1 )
- For 10: ( 10 = 2^1 \times 5^1 )
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Determine the LCM
To find the LCM, we take the highest power of each prime number:- For 2: the highest power is ( 2^2 ) (from 12)
- For 3: the highest power is ( 3^1 ) (from both 15 and 12)
- For 5: the highest power is ( 5^1 ) (from both 15 and 10)
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Construct the LCM
Now, we can find the LCM by multiplying these together: $$ \text{LCM} = 2^2 \times 3^1 \times 5^1 $$ -
Calculate the LCM
Now we compute the product: $$ \text{LCM} = 4 \times 3 \times 5 $$ -
Final Calculation
Calculating step by step:- First, ( 4 \times 3 = 12 )
- Then, ( 12 \times 5 = 60 )
The smallest number that has 15, 12, and 10 as factors is 60.
More Information
The LCM represents the smallest number that is a multiple of all the given factors. In this case, 60 is divisible by 15, 12, and 10, making it the smallest such number.
Tips
- A common mistake is not using the highest power of each prime factor when calculating the LCM. It's important to check all factors accurately.
- Another mistake is forgetting to perform the final multiplication or miscalculating the intermediate steps.
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