¿ìè¶ÌÊÓƵ

#37 Digital multiplication

Set by Mary Ellis

Consider the following rule: Start with any two-digit number, then repeatedly multiply the digits together until the result is a one-digit number. For example: 84 ⇨ 32 ⇨ 6, or 97 ⇨ 63 ⇨ 18 ⇨ 8.

What happens if you start with 93?

There is just one sequence of length 5 starting from a two-digit number. This sequence ends with the number 8. What must the starting number be?

Can you find the longest possible sequence starting from a three-digit number?

Solution next week

#36 Prime generators

Solution

Setting p = 5 generates four primes (5, 7, 11 and 17).

There are three prime numbers between 220 and 230 (223, 227 and 229), and setting p = 227 generates a sequence of four primes (227, 229, 233 and 239).

For x² + x + p, setting x = 0 gives 0² + 0 + p = p and setting x = 1 gives 1² + 1 + p = p + 2, thus generating a twin prime pair (p, p + 2).