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Let's break down the problem of finding the greatest common factor (GCF) of the following terms:

\[
161b^{31}r^7, \quad 14b^{20}r^8, \quad 35b^{30}r^{18}
\]

### Step 1: GCF of the Coefficients (161, 14, 35)
- The prime factorizations of the coefficients are:
  - \( 161 = 7 \times 23 \)
  - \( 14 = 2 \times 7 \)
  - \( 35 = 5 \times 7 \)

  The common factor is 7.

### Step 2: GCF of the \(b\) Terms
- The exponents for \(b\) are 31, 20, and 30. The GCF of these exponents is the smallest value, which is \(b^{20}\).

### Step 3: GCF of the \(r\) Terms
- The exponents for \(r\) are 7, 8, and 18. The GCF of these exponents is the smallest value, which is \(r^7\).

### Step 4: Combine the Results
Now, we combine the GCFs from each step:
\[
\text{GCF} = 7b^{20}r^7
\]

Thus, the greatest common factor (GCF) of the given terms is:

\[
\boxed{7b^{20}r^7}
\]

Would you like any further clarification on this? 

Here are five related questions to consider:
1. How do we find the greatest common factor of two numbers?
2. How does factoring into primes help in finding the GCF?
3. What happens if there are no common factors between the coefficients?
4. How do we apply the GCF concept to polynomials?
5. What is the role of exponents when finding the GCF in algebraic terms?

**Tip:** When finding the GCF of algebraic expressions, always start by factoring the coefficients and then look at the smallest powers of common variables.

Find the greatest common factor (GCF) of the terms 161b^31r^7, 14b^20r^8, and 35b^30r^18.

Learn how to find the greatest common factor (GCF) of algebraic terms 161b^31r^7, 14b^20r^8, and 35b^30r^18. This step-by-step guide covers key algebra concepts such as prime factorization and exponents. Suitable for students in Grades 9-12.

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