AMC 10, Base That Makes a Difference Indivisible by Three

Question

Which integer value of b makes it so that when you subtract the base-bb number 221b ​ from 2021b​, the result isn't a multiple of 3? The choices are:

(A) 3 (B) 4 (C) 6 (D) 7 (E) 8

Answer 1

Subtract 21 in base b: 2021b - 221b to 2000b - 200b. Simplify the expression: 2000b - 200b to 2b^3 - 2b^2. Factor the result: 2b^3 - 2b^2 to 2b^2(b - 1). For non-divisibility by 3, b ≡ 2 (mod 3). Choose b = 8, which matches b ≡ 2 (mod 3).

Answer: (E) 8.