The proper factors of a number n are the factors which are less than n. A number n is deficient if the sum of its proper factors is less than n.
For example, 22 is deficient since 1 + 2 + 11 = 14 < 22.
A number n is super-deficient if twice the sum of its proper factors is less than n. For example, all odd primes p are super-deficient since 2 × 1 = 2 < p.
A number n = pq, where p and q are different odd primes, is not super-deficient. Find all possible values for n. Show that there are no more.
2(p+q) >= pq
if p > 2, then q <= (2p+2)/(p-2)
It is easy to show that if p>11 there is no prime q which satisfies the condition
So, p is either 3,5,7
and you can find the values of q which will work
To solve this problem, we need to find all possible values for the number n, which is the product of two different odd primes (p and q), and show that there are no more values.
First, let's break down the problem into steps:
Step 1: Find the proper factors of n
Step 2: Calculate the sum of the proper factors
Step 3: Check if twice the sum of proper factors is less than n
Step 4: Find all possible values for n by using different odd primes (p and q)
Step 5: Show that there are no more values for n
Step 1: Find the proper factors of n
To find the proper factors of a number n, we need to find all the factors of the number that are less than n.
Step 2: Calculate the sum of proper factors
After finding the proper factors, we need to add them together to find the sum of proper factors.
Step 3: Check if twice the sum of the proper factors is less than n
To determine if a number is super-deficient, we need to check if twice the sum of its proper factors is less than the number itself (2 * sum of proper factors < n).
Step 4: Find all possible values for n using different odd primes (p and q)
We need to find all possible combinations of different odd primes (p and q) and multiply them to get the value for n.
Step 5: Show that there are no more values for n
After finding all possible values for n, we need to demonstrate that there are no more values by showing that any other combination of different odd primes does not meet the given conditions.
By following these steps, we can systematically find the solution to the problem.