Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Mar 02, 2024

Determine if the statement "Any prime number must be deficient" is true or false.

STEP 1

Assumptions
1. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
2. A number is considered deficient if the sum of its proper divisors is less than the number itself.
3. Proper divisors of a number are all the divisors of the number excluding the number itself.

STEP 2

Understand the definition of a deficient number. A number $n$ is deficient if the sum of its proper divisors is less than $n$ .
$\text{Sum of proper divisors of } n < n$

STEP 3

Consider the proper divisors of a prime number. Since a prime number $p$ has no divisors other than 1 and itself, its only proper divisor is 1.

STEP 4

Calculate the sum of proper divisors for a prime number $p$ . Since the only proper divisor is 1, the sum is simply 1.
$\text{Sum of proper divisors of } p = 1$

STEP 5

Compare the sum of proper divisors of a prime number $p$ to the number $p$ itself. Since $p$ is a prime number greater than 1, we have:
$1 < p$

STEP 6

Conclude whether a prime number is deficient. Since the sum of proper divisors (which is 1) is less than the prime number $p$ , every prime number is deficient by definition.
The statement "Any prime number must be deficient" is true.

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