# Show that any composite three-digit number must have a prime factor

• Math100
In summary: Since this is not the case, any composite three-digit number must have a prime factor less than or equal to ## 31 ##.Therefore, we have ## a,b\geq 37 ##.In summary, any composite three-digit number must have a prime factor less than or equal to ## 31 ##.
Math100
Homework Statement
Show that any composite three-digit number must have a prime factor less than or equal to ## 31 ##.
Relevant Equations
None.
Proof:

Suppose for the sake of contradiction that any composite three-digit number
must have a prime factor not less than or equal to ## 31 ##.
Let ## n ## be any composite three-digit number such that ## n=ab ## for
some ## a,b\in\mathbb{Z} ## where ## a,b>1 ##.
Note that the smallest prime factor greater than ## 31 ## is ## 37 ##.
Then we have ## a,b\geq 37 ##.
Thus ## n=ab\geq (37)^2=1369 ##.
This is a contradiction because ## 1369 ## is not a composite three-digit number.
Therefore, any composite three-digit number must have a prime factor less than or equal to ## 31 ##.

Math100 said:
Then we have ## a,b\geq 37 ##.
Better would be "##max(a, b) \ge 37##"
The largest three-digit number is 999. ##\sqrt{999} \approx 31.61##, so if ##n = ab## is a three-digit composite number that is smaller than 999, the larger of its prime factors must be less than or equal to 31.

Math100
I omitted the fact that for any composite, positive integer N, the largest prime factor must be less than or equal to ##\sqrt N##. You might want to prove that as a lemma that supports the problem you're working on.

Mark44 said:
I omitted the fact that for any composite, positive integer N, the largest prime factor must be less than or equal to ##\sqrt N##. You might want to prove that as a lemma that supports the problem you're working on.
Do I need to change my proof to another method? Or is proof by contradiction okay?

I don't see anything wrong with your proof, except that it less direct than it could be.

Mark44 said:
I don't see anything wrong with your proof, except that it less direct than it could be.
About where should I insert the lemma which you've mentioned?

Math100 said:
About where should I insert the lemma which you've mentioned?
A lemma should come before a proof that refers to it.

Lemma: For any composite positive integer ## N ##,
the largest prime factor must be less than or equal to ## \sqrt{N} ##.

Proof:

Note that the largest composite three-digit number is ## 999 ##.
Thus ## \sqrt{N}=\sqrt{999}\approx 31.61 ##.
Therefore, any composite three-digit number must have a prime factor
less than or equal to ## 31 ##.

But you should provide a proof of the lemma...

Math100 said:
Homework Statement:: Show that any composite three-digit number must have a prime factor less than or equal to ## 31 ##.
Relevant Equations:: None.

Proof:

Suppose for the sake of contradiction that any composite three-digit number
must have a prime factor not less than or equal to ## 31 ##.
This is not the negation of the original proposition. The negation should be:

There exists a composite three-digit number with no primes factors less than or equal to 31.

Math100

## 1. What is a composite number?

A composite number is a positive integer that can be divided evenly by at least one number other than 1 and itself. In other words, it has more than two factors.

## 2. Why is it important to prove that any composite three-digit number must have a prime factor?

This proof helps us understand the fundamental properties of numbers and their relationships. It also has practical applications in fields such as cryptography and number theory.

## 3. How do you prove that any composite three-digit number must have a prime factor?

The proof involves using the fundamental theorem of arithmetic, which states that every positive integer can be expressed as a unique product of primes. By examining the possible combinations of three-digit numbers, it can be shown that they all have at least one prime factor.

## 4. Can you provide an example of a composite three-digit number and its prime factor?

One example is the number 210, which can be expressed as the product of 2, 3, 5, and 7. Therefore, its prime factors are 2, 3, 5, and 7.

## 5. Is it possible for a three-digit number to have more than one prime factor?

Yes, it is possible for a three-digit number to have multiple prime factors. For example, the number 210 has four prime factors (2, 3, 5, and 7).

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