91Ó°ÊÓ

Suppose \(a, b,\) and \(c\) are integers. Prove that if \(a \mid b,\) then \(a \mid b c\).

Find the sequence generated by the following generating functions: (a) \(\frac{4 x}{1-x}\). (b) \(\frac{1}{1-4 x}\). (c) \(\frac{x}{1+x}\). (d) \(\frac{3 x}{(1+x)^{2}}\). (e) \(\frac{1+x+x^{2}}{(1-x)^{2}}\) (Hint: multiplication).

Show how you can get the generating function for the triangular numbers in three different ways: (a) Take two derivatives of the generating function for \(1,1,1,1,1, \ldots\) (b) Use differencing. (c) Multiply two known generating functions.

Find the remainder of \(3^{456}\) when divided by (a) 2 . (b) 5. (c) 7 . (d) \(9 .\)

Determine which of the following congruences have solutions, and find any solutions (between 0 and the modulus) by trial and error. (a) \(4 x \equiv 5(\bmod 6)\) (b) \(6 x \equiv 3(\bmod 9)\) (c) \(x^{2} \equiv 2(\bmod 4)\)

Determine which of the following congruences have solutions, and find any solutions (between 0 and the modulus) by trial and error. (a) \(4 x \equiv 5(\bmod 7)\). (b) \(6 x \equiv 4(\bmod 9)\) (c) \(x^{2} \equiv 2(\bmod 7)\).

Solve the following congruence \(5 x+8 \equiv 11(\bmod 22) .\) That is, describe the general solution.

Solve the congruence: \(6 x \equiv 4(\bmod 10)\)

Starting with the generating function for \(1,2,3,4, \ldots,\) find a generating function for each of the following sequences. (a) \(1,0,2,0,3,0,4, \ldots\) (b) \(1,-2,3,-4,5,-6, \ldots\) (c) \(0,3,6,9,12,15,18, \ldots\) (d) \(0,3,9,18,30,45,63, \ldots\) (Hint: relate this sequence to the previous one.)

Solve the congruence: \(341 x \equiv 2941(\bmod 9)\).