Let $\operatorname{GCF}(a,b)$ be the abbreviation for the greatest common factor of $a$ and $b$, and let $\operatorname{LCM}(c, d)$ be the abbreviation for the least common multiple of $c$ and $d$. What is $\operatorname{GCF}(\operatorname{LCM}(8, 14), \operatorname{LCM}(7, 12))$?

We are asked to find $ \operatorname{GCF}(\operatorname{LCM}(8, 14), \operatorname{LCM}(7, 12)) $.

### Step 1: Calculate $ \operatorname{LCM}(8, 14) $
To find the least common multiple (LCM) of 8 and 14, we first find their prime factorizations:

\[
8 = 2^3
\]
\[
14 = 2^1 \times 7^1
\]

The LCM is obtained by taking the highest power of each prime that appears in the factorizations:

\[
\operatorname{LCM}(8, 14) = 2^3 \times 7^1 = 56
\]

### Step 2: Calculate $ \operatorname{LCM}(7, 12) $
Next, we find the LCM of 7 and 12. Their prime factorizations are:

\[
7 = 7^1
\]
\[
12 = 2^2 \times 3^1
\]

Again, we take the highest power of each prime:

\[
\operatorname{LCM}(7, 12) = 2^2 \times 3^1 \times 7^1 = 84
\]

### Step 3: Calculate $ \operatorname{GCF}(56, 84) $
Now, we find the greatest common factor (GCF) of 56 and 84. Start with the prime factorizations:

\[
56 = 2^3 \times 7^1
\]
\[
84 = 2^2 \times 3^1 \times 7^1
\]

The GCF is obtained by taking the lowest power of each common prime factor:

\[
\operatorname{GCF}(56, 84) = 2^2 \times 7^1 = 28
\]

Thus, the answer is:

\[
\boxed{28}
\]

Would you like further details or have any other questions? Here are some related problems to explore:

1. What is the GCF of the LCM of 6 and 15, and the LCM of 9 and 20?
2. Calculate $ \operatorname{GCF}(56, 84, 96) $.
3. Determine the LCM of $ \operatorname{GCF}(18, 24) $ and $ \operatorname{GCF}(30, 45) $.
4. Find $ \operatorname{LCM}(14, 18) $ and $ \operatorname{GCF}(14, 18) $.
5. Calculate $ \operatorname{LCM}( \operatorname{GCF}(56, 84), \operatorname{GCF}(98, 140)) $.

**Tip:** The GCF is found using the smallest exponents in the prime factorization common to all numbers, while the LCM is found using the largest exponents of all primes involved.

Learn how to calculate the greatest common factor (GCF) of the least common multiples (LCM) of numbers 8 and 14, and 7 and 12. This problem covers key concepts in number theory, including GCF, LCM, and prime factorization. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation