Let (a,b)=1 and ab=c^2. Show that a and b are perfect squares.

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In summary, the conversation is regarding a question about showing that two numbers, a and b, are perfect squares when their product is equal to a perfect square, c^2. The conversation also discusses the concept of a and b being relatively prime and suggests using prime factorization to approach the problem. Examples are suggested as a way to think about the problem, but they do not constitute a proof.
  • #1
nikolany
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hi all

I need some help with this question

Let (a,b)=1 and ab=c^2. Show that a and b are perfect squares.

Thank you
 
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  • #2
Look at the prime factorization of a, b and c.
 
  • #3
It took me a little while to comprehend what you meant by (a, b) = 1, but I concluded that you mean that a and b are relatively prime, namely that the largest factor they both share is 1.

To get you thinking about this the right way, try a few examples by assigning different values to a and b to see how that affects their products. The values you choose for a and b should be relatively prime. Keep in mind that your examples won't constitute a proof; they just help you think about the problem, and might help you formulate a real proof.
 

1. How do you define a perfect square?

A perfect square is a number that can be expressed as the product of two equal integers. For example, 9 is a perfect square because it can be written as 3 x 3.

2. What does it mean for two numbers to have a greatest common divisor of 1?

When two numbers have a greatest common divisor of 1, it means that they do not share any common factors other than 1. In other words, they are relatively prime.

3. How does knowing that ab=c^2 help prove that a and b are perfect squares?

Knowing that ab=c^2 allows us to use the fundamental theorem of arithmetic, which states that every positive integer can be expressed as a unique product of prime numbers. Since ab is a perfect square, it can be written as the product of prime numbers raised to even powers. This means that a and b must also be products of prime numbers raised to even powers, making them perfect squares.

4. Can you provide an example to illustrate this concept?

Sure, let's say a = 12, b = 18, and c = 6. We know that (a,b)=1 because the only common factor between 12 and 18 is 1. And we can see that ab = 12 x 18 = 216 = 6^2 x 6^2 = c^2. Therefore, a and b are both perfect squares (6^2 = 36 and 6^2 = 36).

5. Why is it important to show that a and b are perfect squares in this scenario?

This proof is important because it helps us understand the relationship between the greatest common divisor and perfect squares. It also demonstrates the usefulness of the fundamental theorem of arithmetic in solving mathematical problems.

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