Algebra: Proving (a^2,b^2)=1 Using GCD Proof

In summary, (a,b)=1 means that a and b have no common divisors. Using this fact, we can prove that (a^2,b^2)=1 by contradiction. If we assume that (a^2,b^2)=k, where k is greater than 1, then we can show that a^2 and b^2 have a common divisor, contradicting the initial assumption. Therefore, (a^2,b^2)=1.
  • #1
maphec
3
0
(a,b)=d means d is the GCD of a and b

Question:

Let (a,b)=1

Prove: (a^2,b^2)=1



The hint that we were given is to prove this by contradiction ... but, I have no idea how to go about even starting this proof ... Any and all help would be greatly appreciated!
 
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  • #2
Suppose that (a,b) = 1 and that (a2,b2) = k, where k > 1 ...
 
  • #3
Proof:

Supposed n is prime and that n|ab
therefore, n|a or n|b, but not both

Case 1: n|a and n does not divide b
therefore n|a^2
since n does not divide b, n does not divide b^2

Case 2: n|b and n does not divide a
therefore n|b^2
since n does not divide a, n does not divide a^2

Thus, a^2 and b^2 have no common divisors
and therefore gcd(a^2,b^2)=1
 

1. What is the concept of proving (a^2,b^2)=1 using GCD proof?

The concept of proving (a^2,b^2)=1 using GCD proof is based on the properties of the greatest common divisor (GCD) of two numbers. It states that if the GCD of two numbers is equal to 1, then the numbers are relatively prime, meaning they do not have any common factors other than 1. This can be applied to prove that the square of two numbers, a^2 and b^2, is also relatively prime.

2. How does GCD proof work in algebra?

In algebra, GCD proof works by using the definition of the GCD and the properties of multiplication and addition. The GCD of two numbers, a and b, can be expressed as a linear combination of a and b, where one number is multiplied by a constant and the other is multiplied by a different constant. By substituting the values of a^2 and b^2 into this expression, it can be shown that the GCD of a^2 and b^2 is also equal to 1.

3. What is the significance of proving (a^2,b^2)=1 using GCD proof?

Proving (a^2,b^2)=1 using GCD proof is significant because it shows that the two numbers, a and b, do not have any common factors other than 1. This can be applied to various mathematical concepts, such as simplifying fractions, finding the lowest common denominator, and solving equations involving fractions.

4. What are the steps to prove (a^2,b^2)=1 using GCD proof?

The steps to prove (a^2,b^2)=1 using GCD proof are as follows:

  1. Express the GCD of a and b as a linear combination of a and b.
  2. Substitute the values of a^2 and b^2 into the linear combination.
  3. Simplify the resulting expression to show that the GCD of a^2 and b^2 is equal to 1.

5. Can GCD proof be used to prove other concepts in algebra?

Yes, GCD proof can be applied to prove various concepts in algebra. For example, it can be used to show that two fractions are equivalent, or to find the lowest common denominator of two fractions. It can also be used to simplify algebraic expressions and solve equations involving fractions.

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