What is the digit in the units place of 7 raised to the power 73, that is 7^73?

Introduction / Context:
This question asks for the units digit of a very large power, 7^73. Direct multiplication is impossible, so the goal is to use patterns in the units digits of powers of 7. This is a classic modular arithmetic or cyclic pattern problem in number systems.

Given Data / Assumptions:

- We consider 7^73.

- We only need the units digit of this large number.

- No full expansion is required or practical.

Concept / Approach:
The units digits of powers of 7 repeat in a fixed cycle. By computing a few small powers of 7, we can see this repeating pattern and then use the exponent modulo the cycle length to determine the units digit of 7^73. This is a standard cyclicity approach.

Step-by-Step Solution:
Step 1: Compute the first few powers of 7 and record only their units digits. 7^1 = 7 → units digit = 7. 7^2 = 49 → units digit = 9. 7^3 = 343 → units digit = 3. 7^4 = 2401 → units digit = 1. Step 2: Observe the pattern. The units digits go 7, 9, 3, 1. If we continued, 7^5 = 7^4 * 7 would end with 1 * 7 = 7, so the cycle repeats every 4 steps. Step 3: Use the cycle length to find the position of 7^73 in the pattern. The cycle length is 4, so we find 73 modulo 4. 73 / 4 = 18 remainder 1 (because 4 * 18 = 72 and 73 − 72 = 1). So 73 ≡ 1 (mod 4). Step 4: Match the remainder with the pattern. Remainder 1 corresponds to the first element of the cycle, which has units digit 7. Therefore, 7^73 has units digit 7.

Verification / Alternative check:
We know that 7^(4k + 1) always has the same units digit as 7^1, which is 7. Since 73 = 4 * 18 + 1, 7^73 must have the same units digit as 7^1, confirming our earlier reasoning.

Why Other Options Are Wrong:
- 1: This is the units digit for exponents that are multiples of 4 (such as 7^4, 7^8, etc.), not 73.

- 3: Appears as the units digit for exponents congruent to 3 modulo 4 (for example, 7^3, 7^7, ...).

- 9: Appears for exponents congruent to 2 modulo 4 (such as 7^2, 7^6, ...).

Common Pitfalls:
A frequent mistake is to miscompute the cycle length or to incorrectly compute 73 modulo 4. Some students stop after finding only a partial pattern or wrongly assume a different cycle length. Carefully listing the first four powers and using remainder arithmetic avoids these errors.

Final Answer:
The units digit in the expansion of 7^73 is 7.