Find $N$ so that the sequence is the product of three consecutive numbers 
Find the smallest natural number $N$ such that $13 \cdot 17 \cdot N$ is the product of three consecutive natural numbers.

$x(x+1)(x+2) = 13 \cdot 17 \cdot N$.
So let $x=N$, then, $N+1 = 13$ and $N+2 = 17$.
I am unsure how to proceed, can I get a hint?

By finding the gcd of 13 and 17 you can find a solution to $13x-17y=1$. In this case $x=4,y=3$.
Consider $N=3\cdot 4 \cdot 50$
$$13 \cdot 4 =52,17\cdot 3=51$$
$$x=50 \rightarrow x(x+1)(x+2)=50\cdot 51\cdot 52 = 13\cdot 17 \cdot (3\cdot 4\cdot 50)=13\cdot 17 \cdot N$$
Hence smallest value of $N=3\times 4\times 50 = 600$
Solutions for $13x-17y=2$ would be of the form $x=8+13k,y=6+17k$.For $k=-1,x=-5,y=-11$ which is the lowest that we can get for the given case. Hence there doesn't exist any lower solution.
