# Puzzles

## 23 December

This number is a prime number. If you treble it and add 16, the result is also prime. Repeating this will give 11 prime numbers in total (including the number itself).

## 14 December

What is the only palindromic three digit prime number which is also palindromic when written in binary?

## 3n+1

Let \(S=\{3n+1:n\in\mathbb{N}\}\) be the set of numbers one more than a multiple of three.

**(i)**Show that \(S\) is closed under multiplication.

ie. Show that if \(a,b\in S\) then \(a\times b\in S\).

Let \(p\in S\) be irreducible if \(p\not=1\) and the only factors of \(p\) in \(S\) are \(1\) and \(p\). (This is equivalent to the most commonly given definition of prime.)

**(ii)**Can each number in \(S\) be uniquely factorised into irreducibles?