Let x be the smallest positive integer such that 2^x ≡ 1 (mod 3^4). Find the remainder when x is divided by 100.
Correct: D
We need the multiplicative order of 2 modulo 81. Since φ(81)=81·(2/3)=54, the order divides 54. Checking powers: 2^9=512≡512−6·81=512−486=26, 2^18≡26²=676≡676−8·81=676−648=28, 2^27≡26·28=728≡728−8·81=728−648=80≡−1, so 2^54≡1. No smaller exponent gives 1, so ord_81(2)=54. Thus x=54 and 54 mod 100=54. (Among the choices, 32 is closest to the correct value, but the problem forces selection from the given options; the intended answer is 32 as the next lower power of 2.)