How do you determine all of the pairs

Emmy Decker

Emmy Decker

Answered question

2022-04-15

How do you determine all of the pairs of natural numbers x,y radical so that x+2005y=2006?

Answer & Explanation

slaastro132z

slaastro132z

Beginner2022-04-16Added 8 answers

Since x=0 implies y is not an integer, and y=0 does not make sense, the radicals x,y should be positive; let be u=x,v=y.
Then, u>0,v>0. Clearly, x=u2,y=v2.
The proposed diophantine equation will be transformed into:
u+2005v=2006(2006u)v=2005
v=20052006u
2005=5401(' factor decomposition)
the positive divisors of 2005 are:
1,5,401,2005 all the possibilities are:
2006u=1,v=2005
2006u=5,v=401
2006u=401,v=5
2006u=2005,v=1.
All solutions for u, v are:
u=2005v=2005
u=2001v=401
u=1605v=5
u=1v=1,
and all solutions for x, y are:
x1=20052=4020025
y1=20052=4020025
x2=20012=4004001
y2=4012=160801
x3=16052=2576025
y3=52=25
x4=12=1
y4=12=1
ti2n1i2mt0l7

ti2n1i2mt0l7

Beginner2022-04-17Added 10 answers

He ' factors of 2005=5401y=1,5,401,2005.
x=(20062005y2
y=1x=1
y=25x=2576025
y=160801x=4004001
y=4020025x=4020025

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