Math 453: Test 1

Test 1

Tests » Test 1

( 16 points) Complete the following sentences
- Dirichlet's Theorem states…
- For a given integer n, $\nu(n)$ counts…
(15 points) Answer the following questions either “true” or “false.” If true, give a brief justification; if false, give a specific counterexample. In all examples, a,b,c and m are all integers.
- a does not have a multiplicative inverse modulo m if and only if $a \mid m$ .
- If $ca \equiv cb \mod{m}$ then $a \equiv b \mod{m}$ .
- If $(a,m) = (b,m) = 1$ , then ab is relatively prime to m.
(12 points) Is $217 = 7\cdot 31$ a psuedoprime?
(12 points) Express gcd(201,177) as an integral linear combination of 201 and 177.
(12 points) For each of the following linear congruence equations, determine how many incongruent solutions exist. If solutions do exist, provide one solution.
- $177x \equiv 7 \mod{201}$
- $177x \equiv 12 \mod{201}$
(12 points) Solve the following simultaneous system of congruences:

(1)

$\begin{equation*}\begin{split} x&\equiv 3 \mod{7}\ x&\equiv 5 \mod{11}. \end{split}\end{equation*}$

(15 points) How many integers between 1 and 1980
- are divisors of 1980?
- are NOT relatively prime to 1980?
(10 points) Suppose that p and q are a twin prime pair with $3 < p,q$ . Prove that $pq+1$ is a perfect square that is divisible by 9. (Hint: division algorithm with $d=3$ .)