lcm of 14 and 28
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 14 and 28. To solve this, we need to find the smallest positive integer that is a multiple of both numbers.
Answer
$28$
Answer for screen readers
The least common multiple of 14 and 28 is $28$.
Steps to Solve
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Find the prime factorization of each number
First, we need to break down each number into its prime factors.
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For 14: $$ 14 = 2 \times 7 $$
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For 28: $$ 28 = 2^2 \times 7 $$
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Select the highest powers of each prime factor
Now, we look at the prime factors obtained from both numbers and select the highest power for each prime factor.
- The prime factors are 2 and 7.
- The highest power of 2 from both factorizations is $2^2$ (from 28).
- The highest power of 7 from both factorizations is $7^1$.
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Multiply the highest powers together
To find the LCM, we multiply the highest powers of all prime factors:
$$ LCM = 2^2 \times 7^1 = 4 \times 7 $$
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Calculate the result
Now we perform the multiplication:
$$ 4 \times 7 = 28 $$
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Confirm the result
We verify that 28 is a multiple of both 14 and 28:
- $28 \div 14 = 2 \quad (2 \text{ is an integer})$
- $28 \div 28 = 1 \quad (1 \text{ is an integer})$
The least common multiple of 14 and 28 is $28$.
More Information
The LCM of two numbers is the smallest number that both can divide without leaving a remainder. Understanding prime factorization is crucial in finding the LCM efficiently.
Tips
- Confusing the LCM with the greatest common divisor (GCD). Remember that LCM is about multiples, while GCD concerns factors.
- Incorrectly multiplying the numbers instead of using their prime factorizations.
