Asked by Stacey
                :Explain why a prime number must be deficient.
            
            
        Answers
                    Answered by
            MathMate
            
    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 2<sup>n</sup>
= 2<sup>n+1</sup>-1, so all powers of 2<sup>n</sup> 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.
    
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 2<sup>n</sup>
= 2<sup>n+1</sup>-1, so all powers of 2<sup>n</sup> 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.
                                                    There are no AI answers yet. The ability to request AI answers is coming soon!
                                            
                Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.