What is the least common multiple of 21 and 28?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 21 and 28. To find the LCM, we can list the multiples of each number or use the prime factorization method.
Answer
$84$
Answer for screen readers
The least common multiple of 21 and 28 is $84$.
Steps to Solve
- Find the prime factorization of each number
Start by finding the prime factors of 21 and 28.
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For 21, the prime factorization is: $$ 21 = 3 \times 7 $$
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For 28, the prime factorization is: $$ 28 = 2^2 \times 7 $$
- Identify the highest powers of each prime factor
Next, list all unique prime factors from the factorizations and use the highest power of each:
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From 21:
- $3^1$
- $7^1$
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From 28:
- $2^2$
- $7^1$
The unique prime factors are 2, 3, and 7.
- Calculate the least common multiple (LCM)
Multiply the highest powers of each prime factor together:
$$ LCM = 2^2 \times 3^1 \times 7^1 $$
Calculating this, we get:
$$ LCM = 4 \times 3 \times 7 $$
- Final calculation
Now perform the multiplication:
$$ LCM = 4 \times 3 = 12 $$ $$ LCM = 12 \times 7 = 84 $$
So the least common multiple of 21 and 28 is 84.
The least common multiple of 21 and 28 is $84$.
More Information
The least common multiple (LCM) is the smallest positive integer that is a multiple of both numbers. In this case, 84 is not only the LCM but also can be used to find common denominators when adding or subtracting fractions with these numbers.
Tips
- Overlooking prime factors: Sometimes, people forget to check all unique prime factors.
- Miscalculating powers: Be careful with how powers are used in calculations, especially with multiplication.
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