# Puzzles

## Archive

Show me a random puzzle**Most recent collections**

#### Sunday Afternoon Maths LXVII

Coloured weightsNot Roman numerals

#### Advent calendar 2018

#### Sunday Afternoon Maths LXVI

Cryptic crossnumber #2#### Sunday Afternoon Maths LXV

Cryptic crossnumber #1Breaking Chocolate

Square and cube endings

List of all puzzles

## Tags

graphs arrows bases multiplication complex numbers crossnumbers trigonometry cards cryptic clues routes numbers doubling perimeter integration circles calculus addition star numbers division menace speed angles regular shapes quadratics rectangles partitions 2d shapes sequences time 3d shapes christmas surds volume floors triangles probability squares balancing algebra dodecagons shape differentiation folding tube maps coordinates prime numbers wordplay ellipses money spheres shapes parabolas lines ave games sum to infinity palindromes triangle numbers scales indices geometry percentages averages area digits colouring crosswords functions multiples dice rugby remainders means sport polygons square roots chalkdust crossnumber books dates grids cube numbers fractions cryptic crossnumbers advent number factors chocolate mean chess factorials planes integers logic coins irreducible numbers probabilty people maths square numbers sums symmetry perfect numbers proportion odd numbers taxicab geometry unit fractions pascal's triangle hexagons clocks## 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?