Discrete math: proofsFor any three integers x, y, and z, if 𝒚 is divisible by...

Nash Frank

Nash Frank

Answered

2022-07-16

Discrete math: proofs
For any three integers x, y, and z, if 𝒚 is divisible by x and z is divisible by y, then z is divisible by x.
definition: An integer n is divisible by an integer d with D 0, denoted d|n, if and only if there exists an integer k such that n = d k.
Can anyone help, with this problem I don't know how to approach it should I use proof by cases where every number is odd or even. Should I use direct proof or indirect proof?

Answer & Explanation

kitskjeja

kitskjeja

Expert

2022-07-17Added 13 answers

Step 1
If z is divisible by y, then z = a y for some a Z .
Now, y is divisible by x, so y = b x for some b Z .
Step 2
Substituting in the first equation, z = a ( b x ) = ( a b ) x, and a b Z , so z is divisible by x.
Ibrahim Rosales

Ibrahim Rosales

Expert

2022-07-18Added 7 answers

Step 1
If y is divisible by x, we have y = k x for k Z { 0 }. And if z is divisible by y, we have z = m y for m Z { 0 }.
Step 2
Now, putting y = k x to z = m y, we have z = m k x. Therefore by definition, x|z.

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