A one in seven chance
What is the remainder when 2^{164}is divided by 7?
Age
11 to 14
Challenge level
Problem
What is the remainder when $2^{164}$ is divided by 7?
Student Solutions
The remainder when $2^{164}$ is divided by 7 is 4.
Correct solutions came from Alex and Neil of Madras College, St Andrews, who on investigating 2 raised to the power $n$, discovered that the sequence 1,2,4,1,2,4... occurs for increasing values of $n$ which led them to prove their conjectures that:
It was Luke, also of Madras College , who went on to investigate 2
raised to the power $n$, (mod $p$) where $p$ is prime. Extending
the work of Alex and Neil i.e. 2 raised to the power $n$, (mod 7)
which has a period of 3. Luke found that any prime $p$ which can be
written in the form $8k+1$ or $8k-1$ has a period less than or
equal to $(p-1)/2$, this conclusion holding for $k$ belonging to
the set of positive integers and