Your homework is below. Normally, homework is handed in about once a week.

1. Find a counterexample to show that when \(a|b\) and \(c|d\), it is not necessarily true that \(a+c|b+d\).
2. Prove that \(2^n>n\) for all integers \(n\geq 0\) by induction.
3. Write up a proof of the facts we proved in class about the conductor idea with the pairs \(\{2,3\}\), \(\{2,4\}\), and \(\{3,4\}\).
4. Try finding a pattern in the conductors. What is the conductor for \(\{3,5\}\) or \(\{4,5\}\)? Try proving that these work in the same manner as above, and see if you find a pattern in the proof.
5. Color the Foxtrot comic!
6. What is the largest number \(d\) which is a divisor of both 60 and 42?
7. Try to write the answer to the previous problem as \(d=60x+42y\) for some integers \(x\) and \(y\).
8. Get a Sage account at the Gordon Sage server if you don't already have one.