Define D to be the differention operator, i.e.

.

Let's say a polynomial p(x) of degree n is "acceptable" if

where is divisible by 49. It is easy to

verify that is acceptable.

It is an immediate consequence that if p is acceptable, then

so is . If we can show that

is acceptable whenever p is acceptable, we will be done.

So suppose p is an acceptable polynomial of degree n; i.e.

where is divisible

by 49. By the extended rule for differentiation of a product

(which has a binomial-theorem-like look to it),

and , which is divisible

by 49; so is acceptable. We're done.