indirect proof divisor rule?

Good evening,

for the divisor rule

āˆ€š‘Ž,š‘,š‘ āˆˆ Zāˆ¶ š‘Ž | š‘ and š‘Ž āˆ¤ š‘ ā‡’ š‘Ž āˆ¤(š‘+š‘)

Should I use an indirect proof? Unfortunately, I don't know what I should "change" to make it a proof by contradiction.

I would really appreciate any help! Thanks!

(1 votes)
Loading...

Similar Posts

Subscribe
Notify of
3 Answers
Oldest
Newest Most Voted
Inline Feedbacks
View all comments
Rammstein53
11 months ago

Accepted (a | b) and (a | (b+c)), then there is a k,l ā‚¬ Z with

(I) a*k = b

(II) a*l = b + c

(II) is followed by b = a*k:

a*l = a*k + c

a*l – a*k = c

a*(l – k) = c

This follows (a | c), for (l-k) ā‚¬ Z

Objection to (a āˆ¤ c)

  • Option 2:

From a | b follows the existence of a k ā‚¬ Z with a * k = b

From a āˆ¤ c follows the existence of an x ā‚¬ R with a * x = c

This follows

b+c = a*k + a*x

b+c = a*(k + x)

This follows a āˆ¤ (b+c), for (k+x) ā‚¬ R

eterneladam
11 months ago

The opposite is assumed, i.e. a | (b+c).

According to the first condition, a | b applies, it can be concluded directly with the assumption that:

a | (b+c) – b), i.e. a | c,

in opposition to the second condition.