MathDB
1/3 p(n)= s{n)-1, product and sum of digits related

Source: Austrian Polish 1987 APMC

April 30, 2020
sum of digitsproduct of digitsDigitsnumber theory

Problem Statement

For any natural number n=ak...a1a0‾n= \overline{a_k...a_1a_0} (ak≠0)(a_k \ne 0) in decimal system write p(n)=a0⋅a1⋅...⋅akp(n)=a_0 \cdot a_1 \cdot ... \cdot a_k, s(n)=a0+a1+...+aks(n)=a_0+ a_1+ ... + a_k, n∗=a0a1...ak‾n^*= \overline{a_0a_1...a_k}. Consider P={n∣n=n∗,13p(n)=s(n)−1}P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\} and let QQ be the set of numbers in PP with all digits greater than 11. (a) Show that PP is infinite. (b) Show that QQ is finite. (c) Write down all the elements of QQ.
Back to ProblemsView on AoPS