1994 AMC 12 Problem 21

Attempt Problem 21 of the 1994 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1994 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

21.

Find the number of counterexamples to the statement:

“If

N

is an odd positive integer the sum of whose digits is

4

and none of whose digits is

0,

then

N

is prime.”

0

1

2

3

4

A number satisfying the digit conditions has at most four digits

List the odd possibilities by composing

4

into positive digits

The odd possibilities are

13,31,121,211,

and

1111.

The numbers

13,31,

and

211

are prime, while

121=11^2

and

1111=11\cdot101.

Thus there are

2

counterexamples. Thus the correct answer is C.

Problem 21 in other years