AMC 10 Daily Practice Round 3
Complete problem set with solutions and individual problem pages
Call a positive integer if all its digits are nonzero and it is divisible by each of its digits. How many two-digit positive integers are ?
- A.
- B.
- C.
- D.
- E.
Let be a two-digit good integer, where and are nonzero digits. Then, and must both divide . This implies divides and divides . Consider cases on the value of .
Case 1: . Then, the divisibility becomes divides . There are such digits , namely , , and .
Case 2: . Then, the divisibility becomes divides and divides . There are such digits , namely and .
Case 3: . Then, the divisibility becomes divides and divides . There are such digits , namely and .
Case 4: . Then, the divisibility becomes divides and divides . There are such digits , namely and .
Case 5: . Then, the only possible nonzero digit that is a multiple of is itself. Thus, . Then, is guaranteed to divide . Thus, there are good integers here. The total number of two-digit good integers is .
