Understanding the Divisibility of \(97^{30} - 14^{30}\)

The question asks us to determine which number(s) the expression \(97^{30} - 14^{30}\) is divisible by, choosing from 37 and 83.

This expression is in the form of \(a^n - b^n\), where \(a = 97\), \(b = 14\), and \(n = 30\).

There are some important algebraic properties related to the divisibility of expressions in the form \(a^n - b^n\):

• For any positive integer \(n\), the expression \(a^n - b^n\) is always divisible by \(a - b\).
• If \(n\) is an even positive integer, the expression \(a^n - b^n\) is also divisible by \(a + b\).

In our case, \(a = 97\), \(b = 14\), and the exponent \(n = 30\).

Since \(n = 30\) is a positive integer, according to the first property, \(97^{30} - 14^{30}\) must be divisible by \(a - b\).

Let's calculate \(a - b\):

\(a - b = 97 - 14 = 83\)

So, \(97^{30} - 14^{30}\) is divisible by 83.

Next, let's consider the second property. The exponent \(n = 30\) is an even positive integer. Therefore, \(97^{30} - 14^{30}\) must also be divisible by \(a + b\).

Let's calculate \(a + b\):

\(a + b = 97 + 14 = 111\)

So, \(97^{30} - 14^{30}\) is divisible by 111.

Now we need to check if the expression is divisible by 37 and 83 based on the given options. We have already established that it is divisible by 83.

We know the expression is divisible by 111. Let's examine the number 111:

\(111 = 3 \times 37\)

Since 111 is a multiple of 37, any number that is divisible by 111 is also divisible by 37. Because \(97^{30} - 14^{30}\) is divisible by 111, it must also be divisible by 37.

Therefore, the expression \(97^{30} - 14^{30}\) is divisible by both 37 and 83.

Checking Options for Divisibility of \(97^{30} - 14^{30}\)

Let's look at the provided options based on our findings:

• Option 1: 37 but not 83 - This is incorrect because it is divisible by 83.
• Option 2: 83 but not 37 - This is incorrect because it is divisible by 37.
• Option 3: Both 37 and 83 - This matches our conclusion.
• Option 4: Neither 37 nor 83 - This is incorrect as it is divisible by both.

The expression \(97^{30} - 14^{30}\) is divisible by both 37 and 83.

Revision Table: Divisibility Properties

Additional Information: Applying Divisibility Rules

The problem was solved using general algebraic divisibility properties of \(a^n - b^n\). Specifically, when the exponent \(n\) is even, \(a^n - b^n\) is divisible by both \((a-b)\) and \((a+b)\).

In this specific case, \(a=97\) and \(b=14\), with \(n=30\) (an even number). Thus, the expression is divisible by \(97-14=83\) and by \(97+14=111\). Since 111 is a multiple of 37 (\(111 = 3 \times 37\)), any number divisible by 111 is also divisible by 37. Therefore, \(97^{30} - 14^{30}\) is divisible by both 83 and 37.

This method provides a straightforward way to determine divisibility without calculating the large value of \(97^{30} - 14^{30}\).