number property question

[Martoman]

The sum of 5 consecutive numbers is a perfect cube, is a perfect square, find the smallest possible value of r.

These type of question annoy me, as I feel that it has something to do with divisibility, but I really don't understand it that well. If someone could explain it thoroughly, I would appreciate it so I can do divisibility problems with more confidence.

All  I can glean from this is that the sum of 5 consecutive numbers is 5r (cool little property for n consecutive numbers, is n*median but I digress) so

What does that line tell me? I think it means 5 is divisible by which I think means it is divisible by , but still, this line I'm unsure of.

Thanks fellas/gals!

[/0]

That's a nice pickup that 5r = p+q+r+s+t. So we have the eqns:
(btw stop reading anywhere if you wanna try the rest yourself)







Thus we have , .
But since where each , we must have and (the only other possibility is , but then )

However, since the the RHS contains a square , and there is only one 3 on the LHS, the must contain a factor of 3 as well, so

Similarly, since the LHS contains a cube but the RHS has only one 5, there must be two more factors of 5 in the , i.e. . It follows that , since 25 is a square.

In summary, , , , . So and .

We can find more info by letting:



,

And subbing in:





So ,

But this means that , so

We can now check by trial and error that indeed, , .

[Martoman]

A quicker way to do this is to make those preliminary observations then:

since
Then So the smallest

Same as what your calculations verify.