Number Theory
Basic Applications of Modular Arithmetic
There are \(81\) groups of \(21\) students. When we regroup all of the students so that each group has \(5\) members, how many students will be left without a group?
John and Amy have \(48\) and \(51\) quarters, respectively. If they combine all of their quarters and change as much as possible into dollar bills, how many quarters will be left?
Note: There are 4 quarters in a dollar.
November and December have \(30\) and \(31\) days, respectively. If November \(1\) is the first day of the week, how many days of December are there in the last week?
How many different values are there for \( 4 ^ n \pmod { 7} \), as \(n\) ranges over all positive integers?
What is the sum of all positive prime numbers \(x\) such that \(x^2+2\) is also a prime?