7

Part of 1987 Austrian-Polish Competition

Problems(1)

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

Source: Austrian Polish 1987 APMC

For any natural number

n= \overline{a_k...a_1a_0}

(a_k \ne 0)

in decimal system write

p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k

s(n)=a_0+ a_1+ ... + a_k

n^*= \overline{a_0a_1...a_k}

P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}

Q

be the set of numbers in

P

with all digits greater than

1

. (a) Show that

P

is infinite. (b) Show that

Q

is finite. (c) Write down all the elements of

Q

sum of digitsproduct of digitsDigitsnumber theory