SOLUTION: find the gcd of 3+4i and 4+3i in the ring Z[i]

Question 1123424: find the gcd of 3+4i and 4+3i in the ring Z[i]
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
look at the norms
:
N(3+4i) = 9 + 16 = 25
:
N(4+3i) = 16 + 9 = 25
:
the primes dividing each have to divide the rational prime 5, since 5^2 = 25
:
these are (2+i), (2-i)
:
check if (2+i) divides (3+4i)
:
(3+4i)/(2+i) = (3+4i)(2-i)/(2+i)(2-i) = (10+5i)/5
:
so (2+i) divides (3+4i)
:
now check if (2+i) divides (4+3i)
:
(4+3i)/(2+i) = (4+3i)(2-i)/(2+i)(2-i) = (11+2i)/5
:
(2+i) does not divide (4+3i) and it follow that (2-i) divides (4+3i)
:
check if (2-i) divides (3+4i)
:
(3+4i)/(2-i) = (3+4i)(2+i)/(2-i)(2+i) = (2+8i)/5
:
(2-i) does not divide (3+4i)
:
therefore we can conclude that the gcd is 1
:

RELATED QUESTIONS
The lenght of a rectangle is 5cm more than 4 times it's width. The perimeter is 124cm... (answered by stanbon,Alan3354)

I'm on algebra 2 and need help with my work please. It says find the value of the complex (answered by richwmiller)

Find the additive inverse of each number 1. -2 - 3i 2. 1 + 4i 3. 5 - 3i 4. -7 + (answered by drk)

FInd the GCD of 108 and... (answered by user_dude2008)

If z=4-3i and z2=3-4i simplify... (answered by solver91311)

If z = 3 - 4i and w = 8 + 3i,write each expression in the standard form a+bi. A. w - (answered by ikleyn,math_tutor2020)

(3+4i)(4-3i) (answered by Alan3354)

Perform the indicated operation 7. (4-2i)(8+4i) Simplify and write in a+bi form... (answered by lwsshak3)

sketch and label in the complex plane a. (1-2i),(2+3i), (4i), 3 b. The conjugates of... (answered by solver91311)