Problem 1

For a positive integer

m

, denote by

S(m)

and

P(m)

the sum and product, respectively, of the digits of

m

. Show that for each positive integer

n

, there exist positive integers

a_1,a_2,\ldots,a_n

satisfying

S(a_1)<S(a_2)<\cdots<S(a_n)

and

S(a_i)=P(a_{i+1})

for

i=1,2,\ldots,n

, where

a_{n+1}=a_1

Step 3 of 3: Close the cycle

Previous step Next step

a_1:\;2\text{ occurs }k+n-1\text{ times},\quad 1\text{ occurs }2^k-2(k+n-1)\text{ times}

Detailed analysis

Choose $a_1$ with the displayed digit counts; this is possible if $2^k>2(k+n-1)$ . Then $S(a_1)=2^k$ and $P(a_1)=2^{k+n-1}$ . Thus the sums strictly increase, each $S(a_i)$ equals the next product, and $S(a_n)=P(a_1)$ , proving all requirements.