Math 345/645 - Weekly homework 2

Math 345/645 - Weekly homework 2
The problems on this assignment should be written up impeccably and turned in on Tuesday,
February 2. This assignment needs to be typed in LATEX. A problem with a ∗ is extra-credit
for undergraduates and required for graduate students.
1. Prove that if n ≥ 30 is a positive integer, there are non-negative integers x and y so that
n = 6x + 7y.
2. Show that 354321 − 6 is a multiple of 7.
3. ∗ I’m thinking of a polynomial P (x) = ak xk + ak−1 xk−1 + · · · + a0 , where the ai are nonnegative integers for all 0 ≤ i ≤ k. I won’t tell you what the ai are, but you can ask me for
values of P (x). Show that you can determine P (x) uniquely by asking me for only two integer
values of P (x). (That is, you ask to know P (a), and I tell you. Then you can pick some number
b, which can depend on P (a), and I tell you P (b). Then you can determine P (x).)
As an example, determine the unique polynomial with non-negative integer coefficients so that
P (1) = 41 and P (42) = 271100815.