About Factors of numbers?

Question

Question:
Suppose that a and b are integers and that a|b and b|a.
Prove that a=b or a=-b.

My answer to the question:
Since a|b there is an integer z such that b=a*z
Since b|a there is an integer w such that a=b*w
Substitute b=a*z to a=b*w,
a=a*z*w
a/a=z*w
z*w=1

Multiply b=a*z and a=b*w,
a*b=a*z*b*w
a*b=a*b*(z*w)
a*b=a*b*1
a*b=a*b

Then I am stuck from here, I cannot figure out how to prove a=b and a=-b from this point. Can you help me please?

Answer 1

You're almost there.
You've just missed seeing something significant.
You are correct up to z*w = 1.
But z and w are both integers.
The only integers they can be is z = 1 and w = 1,
as 1*1 = 1 is the only possibility.
Thus, b = a*z is b = a*1 = a
and, a = b*w is a = b*1 = b

Answer 2

Since a|b there is an integer z such that b=a*z (by definition of integer division)
Since b|a there is an integer w such that a=b*w (by definition of integer division) Substitute b=a*z to a=b*w,
a=a*z*w
a/a=z*w
Z*w=1
The product of two integers is 1 if and only if the two integers are 1
So z =1 and w=1
So a = b
For the prove a=b or a=-b, at least one of these equations has to be true for the whole statement to be true

Since we have proved for a = b , there is no need for us to prove a = -b (because it’s an or statement) . It therefore means a=b or a=-b.

Answer 3

Falzoon had the right idea, but made a typo.

Z * W = 1. Therefore, Z is a divisor of 1. The only divisors of 1 are 1 and -1.

Therefore, either Z = W = 1 or Z = W = -1. In the first case a = b. In the second case a = -b.

Answer 4

If:
a/b is integer => a >= b ( 'a' greater or equal than 'b')
and
b/a is integer => b >= a ( 'b' greater or equal than 'a')

That is because the result must be an integer.

so, the only solution is.... a=b

Answer 5

why do mathemiticians use shorthand? to bolster superiority? or add to elitedness?
I have a good book here,(infinity in your pocket) and can answer this, BUT i dont understand, "a/b" or "b/a".
do you mean divide? Please be polite, and ask, using language, not slashes, Ie "a" minus "b", or "a" divided by, or "times" by....and i'll tell ya.
imagine its language, would*be confused/say(if)=and then you would=please?
why be confusing.
you can use the update feature to add new content, please do. I will be happy to answer, if you make the notation clearer.

Answer 6

Sounds like a load of gobbelldegook if you ask me.