Hi, I am new to this forum and unfortunately I'm in a number theory class that is kicking my ass. I need some help anywhere I can get it. My homework problem is this:

Prove that for any positive integer k, there exists a sequence of k consecutive positive non-squarefree (squareful?) integers.

So I found this one algorithm, which is kind of similar to the Chinese Remainder Theorem, that says let a1=1, a2=3, a3=5, and list the primes like that. Pick a k.

Then let n1=(a1)^2

Add (a1)^2 repeatedly until you get a number n2 such that

(n2)+1 is congruent to 0 (mod (a2)^2)

Then to this number, add the product ((a1)^2)((a2)^2) repeatedly until you get n3 such that

(n3)+2 is congruent to 0 (mod(a3)^2)

and so on, until you reach nk such that

(nk)+(k-1) is congruent to 0 (mod (ak)^2)

Now, one thing I don't understand is why the numbers in between nk and (nk)+(k-1) are squareful. I know that they are because I've played around with this on the calculator, but I can't show why. For example, I've gotten up to k=4: 14749, 14750, 14751, and 14752. I can see why 14749 and 14752 are squareful, but I don't know why 14750 and 14751 are squareful.

Also, if there is another solution out there, that would be great too!!! Thank you!