Here are the /Solutions.

# 1. An unusual binary operation

For any two rational numbers a and b, define a*b = ab + a + b. Show that the rationals with this operation are a monoid but not a group.

# 2. Square roots

An element x of ℤ^{*}_{m} is called a square root of y mod m if x^{2} = y (mod m). Prove that if m is odd and has at least two distinct prime factors *(was: m is composite, which is not enough)*, then any y in ℤ^{*}_{m} either has no square roots mod m or at least four square roots mod m.

# 3. A big sum

Let p be prime and let a be any integer. Prove that

is a multiple of p if and only if a ≠ 1 (mod p).

# 4. Bicyclic groups

Recall that a group is cyclic if every element can be written as the product of zero or more copies of some single generator g. Call a group *bicyclic*^{1} if every element can be written as the product of some sequence of zero or more copies of two generators g and h (e.g., <>, g, h, gh, hg, g^{2}h^{3}g^{27}hghgh^{2}g, etc.). Prove that S_{n} is bicyclic for any n > 0.

Not a real mathematical term in this context. (1)