ajprincess
  • ajprincess
Please help:) Prove that if a|b and b|a, then \(a=\pm b\)
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Let x=a/b then since b divides a, x is a integer other than 0 Now, Since a also divides b b/a is also integer other than 0 Here, b/a =1/(a/b) =1/x So, x can be either +1 or -1
anonymous
  • anonymous
Now, when x=1 then a/b=1 ===>a=b when x=-1 then a/b=-1 =====>a=-b
anonymous
  • anonymous
Thus, a=+-b

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ajprincess
  • ajprincess
Thanx a lottt @sauravshakya.:)
jiteshmeghwal9
  • jiteshmeghwal9
@sauravshakya i couldn't understand ur statement ``` Now, Since a also divides b b/a is also integer other than 0 ```
jiteshmeghwal9
  • jiteshmeghwal9
@ajprincess can u tell me the statement in easier way ??
ajprincess
  • ajprincess
a|b is a notation that means a divides b and b/a is an integer. b|a is a notation that means b divides a and a/b is an integer.
ajprincess
  • ajprincess
a|b can also be expressed in this way b/a=x, x is an integer. If x is o , b must be equal to 0. b|a can also be expressed in this way a/b=y, y is an integer a/0=infinity. infinity is not an ineger that is y we give the condition that x is an integer except 0. getting it @jiteshmeghwal9?
ParthKohli
  • ParthKohli
Hmm, I take it that you meant a/0 is not defined.
jiteshmeghwal9
  • jiteshmeghwal9
ya! I gt it a little but \(\frac{a}{b}\) doesn't this mean b divides a
ajprincess
  • ajprincess
ya u can take it that way @parthkohli
ajprincess
  • ajprincess
a|b and a/b are different.
ParthKohli
  • ParthKohli
\[a\ne 0, b \ne 0\]because if one of them is zero, then one of \(a|b\) or \(b|a\) will be false. =)
jiteshmeghwal9
  • jiteshmeghwal9
ohh ! i thought it is a/b i gt it completely, thanx :)
ParthKohli
  • ParthKohli
\(b|a\) is the same as \(bx = a :x\in \mathbb{Z}^{+}\)
ajprincess
  • ajprincess
ya u are absolutely right:) @ParthKohli and welcome:) @jiteshmeghwal9
ParthKohli
  • ParthKohli
Nice proof! @sauravshakya

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