Explain why, when a number is composite, its least whole number divisor, other than 1, must be prime.

Choose the correct answer below.

A. If a number [tex]\(d\)[/tex] divides a number [tex]\(n\)[/tex], then every number which divides [tex]\(d\)[/tex] also divides [tex]\(n\)[/tex]. If the least whole number divisor other than 1 was not prime, then it would have a factor other than one and itself, and this factor would also divide [tex]\(n\)[/tex].

B. A composite number can be written as a product of other numbers. Because 2 and 3 are the least positive whole numbers other than 1, and are both prime, the least whole number divisor of a composite number must be prime.

C. A composite number is a number that is the result of a product of primes. Every divisor of a composite number is prime, so the least whole number divisor must also be prime.

D. A composite number has no factors other than 1 and itself, so the least whole number divisor other than 1 would be the number itself, which is prime.

Question

07-07-2024
Mathematics

Answered

Explain why, when a number is composite, its least whole number divisor, other than 1, must be prime.

Choose the correct answer below.

A. If a number [tex]\(d\)[/tex] divides a number [tex]\(n\)[/tex], then every number which divides [tex]\(d\)[/tex] also divides [tex]\(n\)[/tex]. If the least whole number divisor other than 1 was not prime, then it would have a factor other than one and itself, and this factor would also divide [tex]\(n\)[/tex].

B. A composite number can be written as a product of other numbers. Because 2 and 3 are the least positive whole numbers other than 1, and are both prime, the least whole number divisor of a composite number must be prime.

C. A composite number is a number that is the result of a product of primes. Every divisor of a composite number is prime, so the least whole number divisor must also be prime.

D. A composite number has no factors other than 1 and itself, so the least whole number divisor other than 1 would be the number itself, which is prime.

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-07T21:48:35-04:00

Let's carefully analyze why the least whole number divisor of a composite number, other than 1, must be prime.

A composite number is defined as a number that is not prime and has factors other than 1 and itself. For example, 12 is a composite number because it has divisors 1, 2, 3, 4, 6, and 12.

Let's take a closer look at the arguments for each option:

Option A:
- This option states that if a number [tex]\( d \)[/tex] divides a number [tex]\( n \)[/tex], then every number which divides [tex]\( d \)[/tex] also divides [tex]\( n \)[/tex]. If the least whole number divisor other than 1 was not prime, it would have a factor other than 1 and itself, and this factor would also divide [tex]\( n \)[/tex].
- To break this down, let's consider a composite number [tex]\( n \)[/tex]. The least whole number divisor of [tex]\( n \)[/tex] greater than 1 would be a number [tex]\( d \)[/tex] where [tex]\( d \cdot k = n \)[/tex]. If [tex]\( d \)[/tex] is not prime, it would have divisors other than 1 and itself, say [tex]\( p \)[/tex] and [tex]\( q \)[/tex] where [tex]\( p \cdot q = d \)[/tex]. Since [tex]\( d \)[/tex] divides [tex]\( n \)[/tex] and [tex]\( p \)[/tex] divides [tex]\( d \)[/tex], [tex]\( p \)[/tex] must also divide [tex]\( n \)[/tex]. However, this contradicts our assumption that [tex]\( d \)[/tex] is the least whole number divisor greater than 1. Therefore, [tex]\( d \)[/tex] must be prime. This statement is correct.

Option B:
- This option seems to derive its conclusion based on 2 and 3 being the least positive whole numbers other than 1 and being prime. While this is true, it doesn't fully explain why the least whole number divisor of a composite number must be prime.

Option C:
- This option states that every divisor of a composite number is prime. This is incorrect; every divisor of a composite number is not necessarily prime. For example, for the composite number 12, the divisors include 4 and 6, which are not prime.

Option D:
- This option incorrectly states that a composite number has no factors other than 1 and itself. This is the definition of a prime number, not a composite number. Therefore, this option is incorrect.

Therefore, the correct answer is:

OA. If a number [tex]\( d \)[/tex] divides a number [tex]\( n \)[/tex], then every number which divides [tex]\( d \)[/tex] also divides [tex]\( n \)[/tex]. If the least whole number divisor other than 1 was not prime, then it would have a factor other than 1 and itself, and this factor would also divide [tex]\( n \)[/tex].