#7 Home primes
Set by Peter Rowlett
To find the home prime of a number, follow this procedure: first, find its prime factors; then concatenate these by writing the prime factors in size order to form a new number using all of their digits; then repeat until the resulting number is prime.
For example, 9 = 3 脳 3, so its prime factors are 3 and 3. These concatenate to give 33. Now, find the prime factors of 33. They are 3 and 11, which concatenate to give 311. Since 311 is prime, we stop and say that the home prime of 9 is 311.
The home prime of a prime number is the number itself. So, for example, the home prime of 2 is 2.
What is the home prime of 6?
How about the home prime of 10?
Which number less than 20 has a home prime of 1129?
Solution next week
#6 Factor factory
Solution
The factors of 8 are 1, 2, 4 and 8, of which three-quarters are even.
There are 25 numbers between 1 and 100 whose factors are half odd and half even. These are all the numbers that are divisible by 2 but not divisible by 4.
For factors that are divisible by 3, you need the number itself to be divisible by 3. But in that case, at least half of its factors will be divisible by 3 (this is true because for every factor d not divisible by 3, the number 3d will also be a factor), so the answer is zero.
Quick quiz #239
set by Bethan Ackerley
1 What name is given to the retention of juvenile features in adult animals?
2 Enterochromaffin cells are sometimes also called what, after their discoverer?
3 In which country were the oldest known stone tools found?
4 The fluid in the cavities of your elbow and hip joints is known as what?
5 Frank Wilczek and Steven Weinberg are together associated with which hypothetical elementary particle?
Quick quiz #239
Answers
1 Neoteny
2 Kulchitsky cells, after Nikolai Kulchitsky
3 Kenya, at the Lomekwi site
4 Synovial fluid, or synovia
5 The axion