Find the greatest common factor and least common multiple of 48 and 168.
Understand the Problem
The question is asking to find both the greatest common factor (GCF) and the least common multiple (LCM) of the numbers 48 and 168. This involves identifying the factors of both numbers and determining the GCF and LCM based on those factors.
Answer
GCF: $24$, LCM: $336$
Answer for screen readers
GCF: $24$
LCM: $336$
Steps to Solve
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Find the prime factorization of 48
Break down 48 into its prime factors:
$$ 48 = 2^4 \times 3^1 $$ -
Find the prime factorization of 168
Break down 168 into its prime factors:
$$ 168 = 2^3 \times 3^1 \times 7^1 $$ -
Calculate the GCF
The GCF is found by taking the lowest power of each common prime factor:- For $2$: minimum power is $3$ (from $2^3$ in 168)
- For $3$: minimum power is $1$ (shared by both)
Thus, the GCF is:
$$ \text{GCF} = 2^3 \times 3^1 = 8 \times 3 = 24 $$
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Calculate the LCM
The LCM is found by taking the highest power of each prime factor present in either number:- For $2$: maximum power is $4$
- For $3$: maximum power is $1$
- For $7$: maximum power is $1$ (only in 168)
Thus, the LCM is:
$$ \text{LCM} = 2^4 \times 3^1 \times 7^1 = 16 \times 3 \times 7 = 336 $$
GCF: $24$
LCM: $336$
More Information
The GCF and LCM of two numbers are useful in many areas of mathematics. The GCF is used to simplify fractions, while the LCM is often used for finding common denominators in addition or subtraction of fractions. In this case, the GCF of 48 and 168 is 24, and their LCM is 336.
Tips
- Not correctly identifying the prime factors could lead to incorrect calculations.
- Forgetting to take the minimum power for GCF and the maximum power for LCM can result in wrong answers.
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