1994 AIME Problems/Problem 5
Problem
Given a positive integer , let be the product of the non-zero digits of . (If has only one digits, then is equal to that digit.) Let
.
What is the largest prime factor of ?
Solution
Solution 1
Suppose we write each number in the form of a three-digit number (so ), and since our ignores all of the zero-digits, replace all of the s with s. Now note that in the expansion of
we cover every permutation of every product of digits, including the case where that first represents the replaced s. However, since our list does not include , we have to subtract . Thus, our answer is the largest prime factor of .
Solution 2
Note that , and . So , . We add to get 2115. When we add a digit we multiply the sum by that digit. Thus . But we didn't count 100, 200, 300, ..., 900. We add another 45 to get . The largest prime factor of that is .
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.