:Explain why a prime number must be deficient.
A deficient number n is such that the sum of all its divisors is less than 2n.
For example:
16 is deficient because the divisors are:
1,2,4,8,16
But 1+2+4+8+16=31 < 2n
So 16 is deficient.
Since the sum of factors of 2n
= 2n+1-1, so all powers of 2n are deficient.
Similarly, the only factors of a prime number are 1 and n, and since
1+n<2n ∀n>1 (but 1 is not a prime).
Therefore, all primes are deficient.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To understand why a prime number must be deficient, we need to first define what a deficient number is.
A deficient number is a natural number where the sum of its proper divisors (excluding the number itself) is less than the number itself. Proper divisors are the positive divisors of a number, excluding the number itself.
Now, let's consider a prime number P. Since P is a prime number, it has exactly two divisors: 1 and P. Therefore, the sum of its proper divisors is 1. Mathematically, we can represent the sum of the proper divisors of P as:
1 + (P) = 1 + P.
Since the sum of the proper divisors of P (1 + P) is less than the number itself (P), we can conclude that a prime number is indeed deficient.
In summary, a prime number must be deficient because its sum of proper divisors is less than the number itself.
To understand why a prime number must be deficient, we first need to understand the concept of prime numbers and deficient numbers.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number cannot be divided evenly by any other number except 1 and itself.
On the other hand, a deficient number is a natural number that is less than the sum of its proper divisors. Proper divisors are all the positive divisors of a number excluding the number itself.
Now, let's consider a prime number, say p. Since p is a prime number, its only divisors are 1 and itself. Therefore, the sum of its proper divisors is 1.
Since p is a prime number and its proper divisors sum up to 1, which is less than p, we can conclude that p is deficient. In other words, a prime number is always deficient because it is smaller than the sum of its proper divisors.
To summarize, a prime number must be deficient because it has no divisors other than 1 and itself, which means the sum of its proper divisors is always less than the number itself.