What is the unit digit in the expansion of (57242)⁹ˣ⁷ˣ⁵ˣ³ˣ¹ ?
- (a) 2 ✓ UPSC's answer
- (b) 4
- (c) 6
- (d) 8
Why the answer is (a)
• The exponent is 9 × 7 × 5 × 3 × 1 = 945, and the unit digit of the base 57242 is 2.
• Unit digits of powers of 2 repeat in the cycle 2, 4, 8, 6 (period 4).
• 945 = 4 × 236 + 1, so the exponent leaves remainder 1 on division by 4.
• The unit digit is therefore the first in the cycle, 2 — option (a).
Why the other options are wrong
- (b) 4
- Option 4 would be the unit digit if the exponent were congruent to 2 modulo 4, but the exponent 945 is congruent to 1 modulo 4.
- (c) 6
- Option 6 would be the unit digit if the exponent were congruent to 0 modulo 4, but the exponent 945 is not divisible by 4.
- (d) 8
- Option 8 would be the unit digit if the exponent were congruent to 3 modulo 4, but the exponent 945 is congruent to 1 modulo 4.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.