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 .

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  .

kliersel12g · Accepted Answer

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  .Substituing in the first equation,   z  =  a  (  b  x  )  =  (  a  b  )  x, and   a  b  ∈      Z  , so   z is divisible by   x.

Ty Gaines · Accepted Answer

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  }. Now, putting   y  =  k  x to   z  =  m  y, we have   z  =  m  k  x. Therefore by definition,   x      |    z.

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.

Answered question

Answer & Explanation

New Questions in High school geometry