Find all the integer solutions of

1. \(2x+y=2,\)
2. \(3x-4y=0,\)
3. \(15x+18y=17.\)

Find all the solutions in positive integers of

1. \(2x+y=2,\)
2. \(3x-4y=0,\)
3. \(7x+15y=51.\)

Find the different ways a collection of 100 coins (pennies, dimes, and quarters) can be worth exactly $4.99.

Prove or disprove that if \(a \equiv b \pmod{m}\), then \(a^2 \equiv b^2 \pmod{m}\).

Find all \(m\) such that \(1848 \equiv 1914 \pmod{m}\).

Show that every prime (except 2) is congruent to \(1\) or \(3\) \(\pmod{4}\).

Prove or disprove that if \(a \equiv b \pmod{m}\), then \(a^2 \equiv b^2 \pmod{m^2}\).

A man bought some 25-cent stamps, one fourth as many 20-cent stamps, and enough 10-cent stamps to make the total exactly \(n\) dollars. What positive integer values of \(n\) have a unique solution in positive integers?