Question

Please help:)
Prove that if a|b and b|a, then \(a=\pm b\)

Answer 1

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

Answer 2

Now, when x=1 then a/b=1 ===>a=b
when x=-1 then a/b=-1 =====>a=-b

Answer 3

Thus, a=+-b

Answer 4

Thanx a lottt @sauravshakya.:)

Answer 5

@sauravshakya i couldn't understand ur statement

```
Now, Since a also divides b
b/a is also integer other than 0
```

Answer 6

@ajprincess can u tell me the statement in easier way ??

Answer 7

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.

Answer 8

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?

Answer 9

Hmm, I take it that you meant a/0 is not defined.

Answer 10

ya! I gt it a little but \(\frac{a}{b}\) doesn't this mean b divides a

Answer 11

ya u can take it that way @parthkohli

Answer 12

a|b and a/b are different.

Answer 13

\[a\ne 0, b \ne 0\]because if one of them is zero, then one of \(a|b\) or \(b|a\) will be false. =)

Answer 14

ohh ! i thought it is a/b i gt it completely, thanx :)

Answer 15

\(b|a\) is the same as \(bx = a :x\in \mathbb{Z}^{+}\)

Answer 16

ya u are absolutely right:) @ParthKohli and welcome:) @jiteshmeghwal9

Answer 17

Nice proof! @sauravshakya