Find the unit digit of (4137)^754.

Given data

• Base ends with 7: 4137 → units digit behaves like powers of 7.

Concept / Approach

• The units digit of 7^k cycles with period 4: 7, 9, 3, 1, …
• Reduce the exponent modulo 4 to select the cycle position.

Step-by-step calculation

754 ÷ 4 = 188 remainder 2 ⇒ 754 ≡ 2 (mod 4)Cycle position 2 corresponds to units digit 9 (since 7^1→7, 7^2→9).

Verification

7^2 = 49 (units 9), and the cycle repeats every 4 powers; remainder 2 always maps to units 9.

Common pitfalls

• Using the full number instead of just the units digit.
• Forgetting that the cycle length for 7 is 4, not something else.

Final Answer

9