Asked by Stacey
:Explain why a prime number must be deficient.
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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.
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