(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...

Question

Question

(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...

(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every nonzero integers a, b, a, b are relatively prime if and only if a and −b are relatively prime.

math Advanced-Math

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

i) let gcd(a,b) = x

So x divides a==> x |a

similarly x|b

Now Lets take an example gcd(30,22) = 2

So 2|30 and 2|22

now if instead of 30 it is -30,the divisibility doesnt matter with sign

Therfore 2|-30

As x|-b

Therefore gcd(a,b) = gcd(-a,b)

ii) The proof is similar to above

If a, b are relatively prime the gcd(a,b) = 1

we know that gcd(a,b) = gcd(a,-b) (Similar to above)

Therefore gcd(a,-b) = 1

So , a and −b are relatively prime

Add a comment

Answer 2

Similar Homework Help Questions

Let p and n be integers. Prove that, if p is prime, then gcd(p, n) =...

Let p and n be integers. Prove that, if p is prime, then gcd(p, n) = p or gcd(p, n) = 1. . . (i.) Using proof by contrapositive (ii.) Using proof by contradiction
(A) If d=gcd(a,b) and m=lcm(a,b), prove that dm=|ab|. (B) Show that lcm(a,b)=ab if and only if...

(A) If d=gcd(a,b) and m=lcm(a,b), prove that dm=|ab|. (B) Show that lcm(a,b)=ab if and only if gcd(a,b)=1 (C) Prove that gcd(a,c)=gcd(b,c)=1 if and only if gcd(ab,c)=1 for integers a, b, and c. (Abstract Algebra)
correction ---> gcd(a,b) = lcm(a,b) ( Let a and be positive integers. Prove that god(a,b) =...

correction ---> gcd(a,b) = lcm(a,b) ( Let a and be positive integers. Prove that god(a,b) = lama,b) if and only if a

(a) If a | bc, show that a | b*gcd(a,c). (b) If a, b are coprime integers and c | at and c | bt, show that c | t. (c) If...

(a) If a | bc, show that a | b*gcd(a,c). (b) If a, b are coprime integers and c | at and c | bt, show that c | t. (c) If a, b, c are integers with a, c coprime, prove that gcd(ab, c) = gcd(b, c).
If a and b are positive integers, then gcd (a,b) = sa + tb. Prove that...

If a and b are positive integers, then gcd (a,b) = sa + tb. Prove that either s or t is negative.
Prove all non-zero integers a and b, if gcd(a, b) = d then for all non-zero...

Prove all non-zero integers a and b, if gcd(a, b) = d then for all non-zero integers x if a|x and b|x then ab|dx.

EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus,...

EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
3. (i) Prove that the set of all linear combinations of a and b are precisely...

3. (i) Prove that the set of all linear combinations of a and b are precisely the multiples of g.c.d(a,b). (ii)* Prove that a and b are relatively prime iff every integer can be written as a linear combination of a and b.
Exercise 2. Let φ denote the Euler totient function. (i) Prove that for all positive integers...

Exercise 2. Let φ denote the Euler totient function. (i) Prove that for all positive integers m and n, if m,n are relatively prime (coprime), then φ(mn-o(m)o(n) (ii) Is the converse true? Prove or provide a counter-example.

13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...

13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...

(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...

Homework Answers

Add Answer to:
(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...

Post as a guest

Earn Coins

Let p and n be integers. Prove that, if p is prime, then gcd(p, n) =...

(A) If d=gcd(a,b) and m=lcm(a,b), prove that dm=|ab|. (B) Show that lcm(a,b)=ab if and only if...

correction ---> gcd(a,b) = lcm(a,b) ( Let a and be positive integers. Prove that god(a,b) =...

(a) If a | bc, show that a | b*gcd(a,c). (b) If a, b are coprime integers and c | at and c | bt, show that c | t. (c) If...

If a and b are positive integers, then gcd (a,b) = sa + tb. Prove that...

Prove all non-zero integers a and b, if gcd(a, b) = d then for all non-zero...

EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus,...

3. (i) Prove that the set of all linear combinations of a and b are precisely...

Exercise 2. Let φ denote the Euler totient function. (i) Prove that for all positive integers...

13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...

(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...

Homework Answers

Add Answer to: (i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...

Post as a guest

Earn Coins

Add Answer to:
(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every ...