Discrete math: proofsFor 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?

Question

Discrete math: proofsFor 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&#039;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?

kitskjeja · Accepted Answer

Step 1If 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 2Substituting in the first equation,   z  =  a  (  b  x  )  =  (  a  b  )  x, and   a  b  ∈      Z  , so z is divisible by x.

Ibrahim Rosales · Accepted Answer

Step 1If 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 2Now, putting   y  =  k  x to   z  =  m  y, we have   z  =  m  k  x. Therefore by definition, x|z.