- A number \(A\) is a factor of \(B\) if \(B = k \cdot A\) for some integer \(k\).
- Using the property of exponents, express \(7^6 \cdot 11^2\) as \(7 \cdot (7^5 \cdot 11^2)\).
- Since \(7\) is an integer, \(7^5 \cdot 11^2\) is a factor of \(7^6 \cdot 11^2\).
- The correct option is A, as it demonstrates the relationship \(7^6 \cdot 11^2 = 7 \cdot 7^5 \cdot 11^2\).
- The final answer is \(\boxed{A}\).
Explanation
- Problem Analysis and Definition of a Factor
To determine if \(7^5 \cdot 11^2\) is a factor of \(7^6 \cdot 11^2\), we recall that a number \(A\) is a factor of a number \(B\) if there exists an integer \(k\) such that \(B = k \cdot A\). In this problem, we have \(A = 7^5 \cdot 11^2\) and \(B = 7^6 \cdot 11^2\). We aim to find if there is an integer \(k\) such that \(7^6 \cdot 11^2 = k \cdot (7^5 \cdot 11^2)\).
- Simplifying the Expression and Identifying the Factor
Using the exponent rule \(a^{m+n} = a^m \cdot a^n\), we can rewrite the term \(7^6\) as \(7^1 \cdot 7^5\). Substituting this into the expression for \(B\), we get:
\[B = 7^1 \cdot 7^5 \cdot 11^2\]
By the associative property of multiplication, we can group the terms to identify \(A\):
\[B = 7 \cdot (7^5 \cdot 11^2)\]
Comparing this to the definition \(B = k \cdot A\), we see that \(k = 7\).
- Conclusion and Comparison with Options
Since \(k = 7\) is an integer, the condition for \(A\) being a factor of \(B\) is satisfied. Specifically, \(7^6 \cdot 11^2 = 7 \cdot (7^5 \cdot 11^2)\), confirming that \(7^5 \cdot 11^2\) is indeed a factor of \(7^6 \cdot 11^2\). Comparing this result with the provided options:
- A. Yes, because \(7^6 \cdot 11^2 = 7 \cdot 7^5 \cdot 11^2\)
- B. No...
- C. Yes...
- D. No...
Option A is the correct explanation.
Examples
Understanding factors of numbers expressed with exponents is essential in fields like computer science, especially in cryptography and algorithm design. For example, when simplifying large products, checking if one number is a multiple of another by analyzing their prime factorizations (exponents) allows computers to process data much faster. Knowing that \(7^5 \cdot 11^2\) is a factor of \(7^6 \cdot 11^2\) is analogous to simplifying complex code segments into efficient, smaller components.