A Tri-Factorable Positive integer

Question

Found this problem in my SAT book the other day and wanted to see if anyone could help me out.

A positive integer is said to be "tri-factorable" if it is the product of three consecutive integers. How many positive integers less than 1,000 are tri-factorable?

Answer 1

For numbers of this size, you can make a list: $1\times 2\times 3=6, 2\times 3 \times 4= \ldots$ and see where it gets larger than $1000$. A spreadsheet makes it quite easy.

If the upper limit were enough higher, you could let $n$ be the middle number. Then your trifactorable number is $(n-1)n(n+1)=n^3-n$ You can take the cube root of the upper limit and check whether you need one more.

Answer 2

HINT:

$1,000= 10*10*10<10*11*12$ so in the product $n*(n+1)*(n+2)$, $n$ must be less then $10.$