- Compute all square roots of 1,2,3,4,5, 6 (mod 77). Show your work; If
there is no root, explain why.

*(Do***not**use exhaustive search!)

- How many square roots does 1 have modulo 5, 7, 15, 27, 105, and
315? Explain.

- Same as the previous question, but now for cubic (rather than square)
roots.

How many cubic roots does 2 have modulo 3, 5, 7, 15, 105? Compute them.

- Suppose that you have an quadratic equation with roots
*a, b*.

Then this equation can be written as*(x-a)(x-b)=0*: clearly, it has exactly two solutions (*x=a*and*x=b*).

If you open parentheses then you have*x*Use this and the formula for solving quadratic equations to factor^{2 }- (a+b)x +ab=0.

*n*which is a product of two primes, given*n*and*φ(n).*Verify your method by factoring

*n=25651*, given that*φ(n)=25312.*

(No guesswork!)