1994 AIME Problems/Problem 5

Problem

Given a positive integer $n\,$ , let $p(n)\,$ be the product of the non-zero digits of $n\,$ . (If $n\,$ has only one digits, then $p(n)\,$ is equal to that digit.) Let

$S=p(1)+p(2)+p(3)+\cdots+p(999)$

.

What is the largest prime factor of $S\,$ ?

Solution

Note that $p(1)=p(11), p(2)=p(12), p(3)=p(13), \cdots p(19)=p(9)$ , and $p(37)=3p(7)$ . So $p(10)+p(11)+p(12)+\cdots +p(19)=46$ , $p(10)+p(11)+\cdots +p(99)=46*45=2070$ . We add $p(1)+p(2)+p(3)+\cdots +p(10)=45$ to get 2115. When we add a digit we multiply the sum by that digit. Thus $2115\cdot (1+1+2+3+4+5+6+7+8+9)=2115\cdot 46=47\cdot 45\cdot 46$ . But we didn't count 100, 200, 300, ..., 900. We add another 45 to get $45\cdot 2163$ . The largest prime factor of that is $\boxed{103}$ .

1994 AIME (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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1994 AIME Problems/Problem 5

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