2021 AMC 10A Spring Problem 11

Question

For which of the following integers

b

is the base-

b

number

2021_b - 221_b

not divisible by

3?

3

4

6

7

8

Convert

2021_b-221_b

to base

10

Check when

2b^2(b-1)

is divisible by

3

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We can express this expression in base

10

using the definition of bases:

2b^3 + 2b + 1 - 2b^2 - 2b - 1

= 2b^3 - 2b^2 = 2b^2(b - 1).

For this to be divisible by

3,

either

b

or

b - 1

must be divisible by

3.

The only answer choice that satisfies neither of these conditions is

8.

Thus, E is the correct answer.

Problem 11 in other years