What is the least value of n such that the value of \((8^{\mathrm{n}})(25^{24})\) has 49 or more digits?

1. Translate the problem requirements: We need to find the smallest value of n where \((8^n)(25^{24})\) becomes a number with at least 49 digits. A number has 49 digits when it's at least \(10^{48}\) but less than \(10^{49}\).
2. Convert the inequality to powers of 10: Since we want 49 or more digits, we need \((8^n)(25^{24}) \geq 10^{48}\). This gives us a clear target to work toward.
3. Simplify using base relationships: Recognize that \(8 = 2^3\) and \(25 = 5^2\), and that \(2 \times 5 = 10\). This allows us to rewrite our expression in terms of powers of 10 and remaining factors.
4. Restructure to isolate the variable: Rearrange the inequality to solve for n by combining like bases and using the fact that \(10 = 2 \times 5\) to create as many factors of 10 as possible.
5. Solve for the minimum n: Determine the smallest integer value of n that satisfies our inequality, checking against the answer choices to confirm our result.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what it means for a number to have 49 digits. Think about familiar numbers: 1 has 1 digit, 10 has 2 digits, 100 has 3 digits, and so on. We can see that a number has exactly n digits when it's between \(10^{(n-1)}\) and \(10^n - 1\).

For example:

• Numbers with 2 digits: 10, 11, 12, ..., 99 (from \(10^1\) to \(10^2 - 1\))
• Numbers with 3 digits: 100, 101, 102, ..., 999 (from \(10^2\) to \(10^3 - 1\))

So for a number to have 49 or more digits, it must be at least \(10^{48}\).

We need: \((8^n)(25^{24}) \geq 10^{48}\)

Process Skill: TRANSLATE - Converting the digit requirement into a mathematical inequality

2. Convert the inequality to powers of 10

Our goal is clear now. We need to find the smallest value of n where:

\((8^n)(25^{24}) \geq 10^{48}\)

This gives us a concrete target to work toward - we need our expression to be at least as large as \(10^{48}\).

3. Simplify using base relationships

Here's where we use a key insight. Let's break down our numbers into their prime factors:

• \(8 = 2^3\)
• \(25 = 5^2\)
• \(10 = 2 \times 5\)

So our expression becomes:

\((8^n)(25^{24}) = (2^3)^n \times (5^2)^{24} = 2^{(3n)} \times 5^{48}\)

Now here's the clever part: since \(10 = 2 \times 5\), we can create factors of 10 by pairing up 2's and 5's. We have \(5^{48}\), which means we have 48 factors of 5. If we have at least 48 factors of 2, we can make 48 factors of 10.

Process Skill: SIMPLIFY - Breaking complex expressions into manageable prime factors

4. Restructure to isolate the variable

We have \(2^{(3n)} \times 5^{48}\). Let's see how many complete factors of 10 we can make:

If \(3n \geq 48\), then we can write:

\(2^{(3n)} \times 5^{48} = 2^{48} \times 2^{(3n-48)} \times 5^{48} = (2 \times 5)^{48} \times 2^{(3n-48)} = 10^{48} \times 2^{(3n-48)}\)

For our inequality \((8^n)(25^{24}) \geq 10^{48}\) to hold, we need:

\(10^{48} \times 2^{(3n-48)} \geq 10^{48}\)

Dividing both sides by \(10^{48}\):

\(2^{(3n-48)} \geq 1\)

Since \(2^0 = 1\), we need:

\(3n - 48 \geq 0\)

\(3n \geq 48\)

\(n \geq 16\)

5. Solve for the minimum n

From our work above, we found that \(n \geq 16\).

Let's verify this works:

• When \(n = 16\): \(3n = 48\), so \(3n - 48 = 0\)
• This gives us: \(10^{48} \times 2^0 = 10^{48} \times 1 = 10^{48}\)
• So \((8^{16})(25^{24}) = 10^{48}\), which has exactly 49 digits ✓

Let's check \(n = 15\) to confirm it's too small:

• When \(n = 15\): \(3n = 45\), so we get \(2^{45} \times 5^{48}\)
• We can only make 45 complete factors of 10: \(10^{45} \times 5^3 = 10^{45} \times 125\)
• This gives us a number between \(10^{47}\) and \(10^{48}\), which has only 48 digits

Therefore, the minimum value is \(n = 16\).

Final Answer

The least value of n such that \((8^n)(25^{24})\) has 49 or more digits is 16.

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the digit count requirement: Students often confuse "49 or more digits" with "exactly 49 digits" or "more than 49 digits." This leads them to set up the wrong inequality - using > instead of ≥, or aiming for \(10^{49}\) instead of \(10^{48}\) as the threshold.

2. Incorrect digit-to-power relationship: Students may incorrectly think that a number with n digits must be at least \(10^n\) instead of \(10^{(n-1)}\). For example, thinking 100 (3 digits) corresponds to \(10^3\) rather than being \(\geq 10^2\). This shifts their entire target by one power of 10.

3. Missing the prime factorization insight: Students might try to compute \((8^n)(25^{24})\) directly or convert everything to the same base without recognizing that factoring into 2's and 5's allows efficient pairing to create powers of 10. This leads to much more complex calculations.

Errors while executing the approach

1. Incorrect exponent manipulation: When converting \(8^n\) to \((2^3)^n\), students often write \(2^{(n+3)}\) instead of \(2^{(3n)}\), or when converting \(25^{24}\) to \((5^2)^{24}\), they write \(5^{(24+2)}\) instead of \(5^{48}\). These exponent rule errors completely derail the solution.

2. Mishandling the pairing of 2's and 5's: Students may incorrectly assume they can always make the maximum number of 10's without checking which prime factor is limiting. For instance, if they have \(2^{45} \times 5^{48}\), they might incorrectly think they can make 48 factors of 10 instead of only 45.

3. Algebraic errors in the inequality: When solving \(3n \geq 48\), students might make basic algebraic mistakes like forgetting to divide by 3, or incorrectly handling the inequality direction, leading to wrong values like \(n \geq 48\) or \(n \geq 144\).

Errors while selecting the answer

1. Boundary condition confusion: Students might find \(n \geq 16\) but then select \(n = 15\), thinking they need the largest value that's less than 16, rather than understanding they need the smallest value that satisfies the inequality.

2. Verification errors: Students may correctly find \(n = 16\) but then incorrectly verify their answer, making computational errors that cause them to doubt their result and select a different answer choice instead.