The digit in the unit's place of the product 3^1991 is _________. A. 7 B. 1 C. 3 D. 9

GATE 2023 · General Aptitude · Modular Arithmetic · medium

Answer: The digit in the unit's place of 3^1991 is 7 (option A).

1. Write out the units digit cycle for powers of 3: 3^1 = 3 (units: 3); 3^2 = 9 (units: 9); 3^3 = 27 (units: 7); 3^4 = 81 (units: 1); 3^5 = 243 (units: 3) — cycle restarts. The cycle is {3, 9, 7, 1} with period 4.
2. Find 1991 mod 4: 4 * 497 = 1988; 1991 - 1988 = 3. So 1991 mod 4 = 3. The exponent 1991 corresponds to position 3 in the cycle {3, 9, 7, 1}, which is 7.