Is $fft^{-1}(fft(x)\times fft(y))$ integer?

Question

I have some troubles with the following question:

Let ($x_0, x_1, ..., x_{N-1}$) and ($y_0, y_1, ..., y_{N-1}$) be two sequences of integers from the set {$0,1,...,9$}. Let ($p_0, p_1, ..., p_{N-1}$) be the sequence defined by $p = fft^{-1}(fft(x)\times fft(y))$. Prove that all $p_i$ are integers and that $0 \leq p_i \leq 100N$.

I have tried to solve it using the definition of the fourier coefficients but I end up with a mess of products and sums... I hope you can help me. Thank you very much in advance!

Answer 1

$ifft(X \times Y)=ifft(X) * ifft(Y)$ with * a convolutive product. Consequently $p = x * x$ (considering $x$ is repeated such that x is period N ($x(i+N)=x(i)$).

So, $p_i=\sum_{j=1}^N x_jx_{i-j} \leq \sum_{j=1}^N 9\times9 = N81$ And $p_i$ is a sum and product of integer so it is interger.