Puzzles

Arrange the digits 1–9 (using each digit exactly once) so that the three digit number in: the middle row is a prime number; the bottom row is a square number; the left column is a cube number; the middle column is an odd number; the right column is a multiple of 11. The 3-digit number in the first row is today's number.

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).

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

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?