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## Sunday, 14 September 2014

### Sunday Afternoon Maths XXVIII

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

#### 2009

2009 unit cubes are glued together to form a cuboid. A pack, containing 2009 stickers, is opened, and there are enough stickers to place 1 sticker on each exposed face of each unit cube.
How many stickers from the pack are left?

#### 3$$n$$+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?