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 4 of 7: Factor the only possible equality

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x+yz=(y+z)x\Longleftrightarrow (y-x)(z-x)=x(x-1)

Detailed analysis

Expanding and rearranging the possible equality gives $yz-xy-xz+x=0$ . Factoring yields $(y-x)(z-x)=x(x-1)$ , which is the exact criterion for repeated combination values.