:Explain why a prime number must be deficient.

Answer 1

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.