Thread: Digit sum & digit product of number x

For every natural number x, let S(x) be the sum and P(x) the product of
the (decimal) digits of x. Show that for each natural number n there exist infinitely
many values of x such that.

The problem is from final round of Austrian MO '83 and I really can't find any solution on the internet. So I am asking for help, maybe you know it, thanks in advance.

Originally Posted by fhactor

For every natural number x, let S(x) be the sum and P(x) the product of
the (decimal) digits of x. Show that for each natural number n there exist infinitely
many values of x such that.

The problem is from final round of Austrian MO '83 and I really can't find any solution on the internet. So I am asking for help, maybe you know it, thanks in advance.

Here's one suggestion. Others may have neater constructions.

For positive integers p,q, let be the integer whose decimal form consists of p 1s followed by q 0s. In other words, . Then and . Now take . Then , and . As q varies, that gives an infinite family of integers with .