2003 AMC 10B Problem 25

Question

How many distinct four-digit numbers are divisible by

3

and have

23

as their last two digits?

27

30

33

81

90

A number is divisible by

3

exactly when its digit sum is divisible by

3

Writing the number as

\overline{ab23},

require

a+b+5

divisible by

3,

so

a+b\equiv 1 \pmod 3

Write the number as

\overline{ab23}.

It is divisible by

3

when

a+b+2+3=a+b+5

is divisible by

3,

that is, when

a+b\equiv 1 \pmod 3.

The two-digit prefix

\overline{ab}

ranges over the

90

values from

10

to

99,

and exactly one third of them satisfy this, giving

\dfrac{90}{3}=30.

Thus, the correct answer is B.

Problem 25 in other years