Problem 1
Find the prime-power decomposition of \(2907\).
Practice Final Examination
Problem 2
Find the least residue of \(23472\) modulo \(245\).
Problem 3
Find all integer solutions to the equation \(492x+160y=16\).
Problem 4
Determine the least residue of \(203^5-2\cdot 113^3+4\) modulo \(20\).
Problem 5
Find all solutions to \(20x \equiv 8 \pmod{44}\).
Problem 6
Find all integer solutions to the system of congruences: \[\begin{align*} x &\equiv 3 \pmod{4}\\ 5x &\equiv 3 \pmod{8}\\ 2x &\equiv 3 \pmod{13}. \end{align*}\]
Problem 7
What is the remainder when \(17^{41}\) is divided by \(50\)?
Problem 8
Compute \(\sigma(1250)\).
Problem 9
Compute \(\phi(12672)\).
Problem 10
Determine all primitive roots modulo \(14\).
Problem 11
Determine whether \(x^2 = 7 \pmod{31}\) has a solution.
Problem 12
Compute the Legendre symbol \(\left(\frac{97}{131} \right)\).
Problem 13
Suppose \(f(n)\) is a function of a natural number \(n\) such that \(f(0)=1\) and \(f(n+2)=(n+2)f(n)\). Prove that \(f(2n)=2^n n!\).
Problem 14
Show that the difference of two consecutive cubes is never divisible by \(5\).
Problem 15
Prove that if \(r! \equiv (-1)^r \pmod{p}\), then \((p-r-1)! \equiv -1 \pmod{p}\).
Problem 16
Prove that \(\phi(n)=\frac{n}{3}\) if and only if \(n\) is divisible by both \(2\) and \(3\), but no other primes.