If x is a positive integer, which of the following CANNOT be the units digit of 7^(x)?

1. Translate the problem requirements: We need to find which units digit among the choices (1, 3, 5, 7, 9) can NEVER appear as the units digit when 7 is raised to any positive integer power.
2. Identify the units digit pattern for powers of 7: Calculate the first few powers of 7 and observe only their units digits to find the repeating cycle.
3. Determine the complete cycle: Continue calculating until the units digit pattern repeats, establishing all possible units digits for \(7^x\).
4. Compare with answer choices: Check which answer choice does not appear in our established cycle of possible units digits.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this question is really asking us. We have the number 7, and we're raising it to different positive integer powers like \(7^1\), \(7^2\), \(7^3\), and so on. Each time we calculate one of these powers, we get a number that ends in some digit - that's called the "units digit." For example, if \(7^3 = 343\), then the units digit is 3.

The question wants to know: out of the five choices given (1, 3, 5, 7, 9), which one can NEVER appear as the units digit when we calculate \(7^x\) for any positive integer x.

Process Skill: TRANSLATE - Converting the problem language "which of the following CANNOT be" into the mathematical task of finding excluded values from a pattern.

2. Identify the units digit pattern for powers of 7

Let's calculate some powers of 7 and see what happens to the units digits. We don't need to calculate the full numbers - we only care about what digit appears at the end:

\(7^1 = 7\) → units digit is 7
\(7^2 = 49\) → units digit is 9
\(7^3 = 343\) → units digit is 3
\(7^4 = 2,401\) → units digit is 1

So far, we've seen the units digits: 7, 9, 3, 1. Let's continue:

\(7^5 = 16,807\) → units digit is 7

Wait! Notice that \(7^5\) has the same units digit as \(7^1\). This suggests the pattern might be starting to repeat.

3. Determine the complete cycle

Since \(7^5\) has the same units digit as \(7^1\), let's check if the pattern truly repeats:

\(7^5\) ends in 7 (same as \(7^1\))
\(7^6 = 7^5 \times 7\), and since \(7^5\) ends in 7, we get \(7 \times 7 = 49\), so \(7^6\) ends in 9 (same as \(7^2\))
\(7^7\) would end in 3 (same as \(7^3\))
\(7^8\) would end in 1 (same as \(7^4\))

Perfect! The pattern repeats every 4 powers. The complete cycle of units digits for powers of 7 is:

Position in cycle: 1st → 2nd → 3rd → 4th → (repeats)
Units digit: 7 → 9 → 3 → 1 → (back to 7)

This means the only possible units digits for \(7^x\) are: 7, 9, 3, and 1.

4. Compare with answer choices

Now let's check each answer choice against our discovered pattern:

1. 1 - This appears in our cycle (\(7^4\), \(7^8\), \(7^{12}\), etc.)
2. 3 - This appears in our cycle (\(7^3\), \(7^7\), \(7^{11}\), etc.)
3. 5 - This does NOT appear anywhere in our cycle!
4. 7 - This appears in our cycle (\(7^1\), \(7^5\), \(7^9\), etc.)
5. 9 - This appears in our cycle (\(7^2\), \(7^6\), \(7^{10}\), etc.)

Process Skill: CONSIDER ALL CASES - We systematically checked every answer choice against our complete pattern to ensure we didn't miss any possibilities.

Final Answer

The units digit 5 never appears in any power of 7. Our cycle shows that powers of 7 can only end in 1, 3, 7, or 9. Since 5 is not part of this repeating pattern, it CANNOT be the units digit of \(7^x\) for any positive integer x.

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "CANNOT be" means

Students often confuse this with "which of the following IS the units digit" and look for what appears in the pattern rather than what's missing. They need to recognize that they're looking for the one digit that NEVER appears in any power of 7.

2. Not recognizing the need to find a complete cycle

Some students calculate just a few powers (like \(7^1\), \(7^2\), \(7^3\)) and assume they've seen all possible units digits. They don't realize that units digit patterns for powers are cyclical and they need to find the complete repeating cycle to be certain they haven't missed any possibilities.

Errors while executing the approach

1. Arithmetic errors in calculating powers

When calculating \(7^2\), \(7^3\), \(7^4\), etc., students may make multiplication errors. For example, incorrectly calculating \(7^3\) as \(7^2 \times 7 = 49 \times 7\) and getting the wrong units digit, which would throw off their entire pattern.

2. Not properly tracking when the cycle repeats

Students might continue calculating powers beyond where the pattern repeats (after \(7^4\)) without recognizing that \(7^5\) gives the same units digit as \(7^1\). They may think they need to calculate many more powers or miss that the cycle has already completed.

3. Focusing only on units digits during multiplication

While the shortcut of only considering units digits (like \(7 \times 7 = 49\), so units digit is 9) is valid, students sometimes apply this incorrectly or inconsistently, leading to errors in their pattern identification.

Errors while selecting the answer

1. Selecting a digit that DOES appear in the cycle

After correctly identifying the cycle as 7→9→3→1, students might accidentally select an answer choice like A(1) or D(7) because these digits stood out to them, forgetting that the question asks for what CANNOT be the units digit.