Problem 3

Let

n>3

be an integer. Choose three numbers from

\{1,2,\ldots,n\}

. Using each once, together with addition, multiplication, and parentheses, form all possible combinations. (a) Show that if all three chosen numbers are greater than

n/2

, their values are all distinct. (b) Let

p

be a prime with

p\le \sqrt{n}

. Show that the number of choices whose smallest number is

p

and whose combination values are not all distinct is exactly the number of positive divisors of

p-1

Step 1 of 7: Name the chosen numbers and list the combinations

Previous step Next step

1<x<y<z,\qquad x+y+z,\ x+yz,\ xy+z,\ y+zx,\ (x+y)z,\ (z+x)y,\ (y+z)x,\ xyz

Detailed analysis

The smallest chosen number is at least $2$ in both parts: in (a), $n/2>1$ , and in (b), $p$ is prime. Write the three chosen numbers as $1<x<y<z$ . Up to commutativity, the eight possible values are the expressions displayed above.