2013 AMC 10A Problem 13

Question

How many three-digit numbers are not divisible by

5,

have digits that sum to less than

20,

and have the first digit equal to the third digit?

52

60

66

68

70

Answer: B

Concepts:digits casework

Difficulty rating: 1370

Small hint:

The number has form

aba

Big hint:

Case on the repeated digit

a

, excluding values that make the number divisible by

5

Solution:

Note that for the number to not be divisible by

5,

the units digits cannot be either

0

or

5.

Let

x

be the hundreds and units digit and

y

be the tens digit. Then we want

2x + y \lt 20.

Casing on the

8

possible values of

x,

we get:

If

x

is

1, 2, 3,

or

4,

then

y

can be anything since

y \lt 10.

If

x = 6,

then

y \lt 8,

which gives us

8

solutions.

If

x = 7,

then

y \lt 6,

which gives us

6

solutions.

If

x = 8,

then

y \lt 4,

which gives us

4

solutions.

If

x = 9,

then

y \lt 2,

which gives us

2

solutions.

This gives us a total of

4 \cdot 10 + 8 + 6 + 4 + 2 = 60

solutions.

Thus, B is the correct answer.

2013 AMC 10A Problem 13

Answer: B

Difficulty rating: 1370

Small hint:

Big hint:

Solution:

Problem 13 in other years