Which of the following is always a factor of every prime number

Mathematics
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which of the following is always a factor of every prime number

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Which of the following is always a factor of every prime number?

Answer:

To determine which factor is always present in every prime number, let’s start by understanding what a prime number is.

Prime Numbers: Definition and Characteristics

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In simpler terms, a prime number has exactly two distinct positive divisors: 1 and the number itself. For example, the number 7 is prime because its only divisors are 1 and 7.

Step-by-Step Analysis

1. Characteristics of a Prime Number:

  • A prime number ( n ) has only two divisors: 1 and ( n ).
  • The divisors of a prime number do not include any numbers other than 1 and itself.

2. Common Divisors of Prime Numbers:

  • Since every prime number ( p ) has the divisors 1 and ( p ), we must take into account which of these is common to all primes.
  • By examining the definition, it is clear that 1 will always be a divisor of any prime number because:
    • 1 divides all integers.

3. Conclusion:

The number 1 is always a factor of every prime number.

Further Explanation

This characteristic of prime numbers is fundamental to their definition. Let’s delve deeper into why only 1 and the prime number itself are considered its divisors:

Divisibility Check:

  • Division Concept: For any integer ( a ), if ( a ) divides another integer ( b ) without leaving a remainder, then ( a ) is said to be a divisor of ( b ). Mathematically, if ( b \div a = k ) (where ( k ) is an integer with no remainder), then ( a \mid b ).
  • Prime Number Example: Let’s use 17 as an example.
    • Dividing 17 by 1 gives exactly 17, which means 1 is a divisor.
    • Dividing 17 by 17 results in 1, which also confirms 17 as its own divisor.
    • Trying any number between 1 and 17 (other than 17 itself) does not give an integer result, reinforcing 1 and 17 as the only divisors of 17.

Inverse of Primality:

  • Composite Numbers: A composite number, unlike a prime, can be divided evenly by numbers other than 1 and itself. For example, 12 divides by 1, 2, 3, 4, 6, and 12.
  • Importance of 1: The number 1 is a unique factor in that it divides every integer, prime or composite, without exception. This fundamental property establishes why 1 is necessary in the definition of prime numbers.

Summary of Key Points:

  • Prime Characterization: Prime numbers are those greater than 1; definitionally, they can only be divided completely (no remainder) by 1 or themselves.
  • Universal Factor: The integer 1 is universally applicable as a divisor to both prime and composite numbers, retaining its status as a basic unit of divisibility in mathematics.

In summary, the key takeaway is that every prime number, without exception, includes the number 1 as one of its factors. Understanding this naturally follows from the properties that define primality, setting them apart from composite numbers in terms of divisor count and simplicity.