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?

Show answer & extension


Show me a random puzzle
 Most recent collections 

Advent calendar 2020

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

List of all puzzles


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


Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2021