Question

Let a, b, c be natural numbers. We say that (a, b, c) is a Pythagorean...

Let a, b, c be natural numbers. We say that (a, b, c) is a Pythagorean triple, if a2 + b2 = c2 . For example, (3, 4, 5) is a Pythagorean triple. For the next exercises, assume that (a, b, c) is a Pythagorean triple.

(c) Prove that 4|ab Hint: use the previous result, and a proof by con- tradiction.
(d) Prove that 3|ab. Hint: use a proof by contradiction.
(e) Prove that 12 |ab. Hint : Use the two previous results, plus a theorem
proved in class.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Proof by contradiction: Suppose a right triangle has side lengths a, b, c that are natural...
Proof by contradiction: Suppose a right triangle has side lengths a, b, c that are natural numbers. Prove that at least one of a, b, or c must be even. (Hint: Use Pythagorean Theorem)
In class we proved that if (x, y, z) is a primitive Pythagorean triple, then (switching...
In class we proved that if (x, y, z) is a primitive Pythagorean triple, then (switching x and y if necessary) it must be that (x, y, z) = (m2 − n 2 , 2mn, m2 + n 2 ) for some positive integers m and n satisfying m > n, gcd(m, n) = 1, and either m or n is even. In this question you will prove that the converse is true: if m and n are integers satisfying...
Prove: Let (a,b,c) be a primitive pythagorean triple. then we have the following 1. gcd(c-b, c+b)...
Prove: Let (a,b,c) be a primitive pythagorean triple. then we have the following 1. gcd(c-b, c+b) =1 2. c-b and c+b are squares
4. Let a, b, c be integers. (a) Prove if gcd(ab, c) = 1, then gcd(a,...
4. Let a, b, c be integers. (a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) = 1. (Hint: use the GCD characterization theorem.) (b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) = 1. (Hint: you can use the GCD characterization theorem again but you may need to multiply equations.) (c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if and...
Prove: If (a,b,c) is a primitive Pythagorean triple, then either a or b is divisible by...
Prove: If (a,b,c) is a primitive Pythagorean triple, then either a or b is divisible by 3.
1. Prove that an integer a is divisible by 5 if and only if a2 is...
1. Prove that an integer a is divisible by 5 if and only if a2 is divisible by 5. 2. Deduce that 98765432 is not a perfect square. Hint: You can use any theorem/proposition or whatever was proved in class. 3. Prove that for all integers n,a,b and c, if n | (a−b) and n | (b−c) then n | (a−c). 4. Prove that for any two consecutive integers, n and n + 1 we have that gcd(n,n + 1)...
Let p and q be two real numbers with p > 0. Show that the equation...
Let p and q be two real numbers with p > 0. Show that the equation x^3 + px +q= 0 has exactly one real solution. (Hint: Show that f'(x) is not 0 for any real x and then use Rolle's theorem to prove the statement by contradiction)
Let (a, b, c, d) be elements within N+ (all positive natural numbers). If a is...
Let (a, b, c, d) be elements within N+ (all positive natural numbers). If a is less than/equal to b, and c is less than/equal to d, then (a*c) is less than/equal to (b*d) with equality if and only if a=b and c=d. Proof?
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all i}. It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all i}. It turns out that R forms a ring under the operations: (a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···), (a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···) Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}. You may use without proof the fact that I forms an ideal of R. a) Is I principal in R? Prove your claim. b) Is I prime in R? Prove your claim....
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all i}. It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all i}. It turns out that R forms a ring under the operations: (a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···), (a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···) Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}. You may use without proof the fact that I forms an ideal of R. a) Is I principal in R? Prove your claim. b) Is I prime in R? Prove your claim....
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT