Theorem: If w, x, y, z are integers where w divides x and y divides z, then wy divides xz. Proof. Let w, x, y, z be integers such that w divides x and y divides z. Since w divides x, then x = kw and w # 0. Since y divides z, then z = jy and y#0. Plug in the expression kw for x and jy for z in the expression xz to get x2 = (kw) (jy) = (kj) (wy) Since k and j are integers, then kj is also an integer. Since w #0 and y # 0, then wy # 0. Since xz equals wy times an integer and wy #0, then wy divides xz. ■ *There is something wrong with the proof, so please do not tell me that the proof is actually correct. Thank you.
Theorem: If w, x, y, z are integers where w divides x and y divides z, then wy divides xz. Proof. Let w, x, y, z be integers such that w divides x and y divides z. Since w divides x, then x = kw and w # 0. Since y divides z, then z = jy and y#0. Plug in the expression kw for x and jy for z in the expression xz to get x2 = (kw) (jy) = (kj) (wy) Since k and j are integers, then kj is also an integer. Since w #0 and y # 0, then wy # 0. Since xz equals wy times an integer and wy #0, then wy divides xz. ■ *There is something wrong with the proof, so please do not tell me that the proof is actually correct. Thank you.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hello. Please answer the attached Discrete Mathematics question correctly and follow all directions. Correctly explain where the proof uses invalid reasoning or skips essential steps.
*If you would like for me to give you a thumbs up then just answer the question correctly by yourself and do not copy from outside sources. Thank you.
![**Theorem:** If \( w, x, y, z \) are integers where \( w \) divides \( x \) and \( y \) divides \( z \), then \( wy \) divides \( xz \).
**Proof.**
Let \( w, x, y, z \) be integers such that \( w \) divides \( x \) and \( y \) divides \( z \). Since \( w \) divides \( x \), then \( x = kw \) and \( w \neq 0 \). Since \( y \) divides \( z \), then \( z = jy \) and \( y \neq 0 \). Plug in the expression \( kw \) for \( x \) and \( jy \) for \( z \) in the expression \( xz \) to get
\[
xz = (kw)(jy) = (kj)(wy)
\]
Since \( k \) and \( j \) are integers, then \( kj \) is also an integer. Since \( w \neq 0 \) and \( y \neq 0 \), then \( wy \neq 0 \). Since \( xz \) equals \( wy \) times an integer and \( wy \neq 0 \), then \( wy \) divides \( xz \). ∎
*There is something wrong with the proof, so please do not tell me that the proof is actually correct. Thank you.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1223c4c6-4ebb-4911-bc0d-edb37c26385f%2F5f3610b8-e73d-4f01-9b99-b395d4336ac1%2F85hldcj_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem:** If \( w, x, y, z \) are integers where \( w \) divides \( x \) and \( y \) divides \( z \), then \( wy \) divides \( xz \).
**Proof.**
Let \( w, x, y, z \) be integers such that \( w \) divides \( x \) and \( y \) divides \( z \). Since \( w \) divides \( x \), then \( x = kw \) and \( w \neq 0 \). Since \( y \) divides \( z \), then \( z = jy \) and \( y \neq 0 \). Plug in the expression \( kw \) for \( x \) and \( jy \) for \( z \) in the expression \( xz \) to get
\[
xz = (kw)(jy) = (kj)(wy)
\]
Since \( k \) and \( j \) are integers, then \( kj \) is also an integer. Since \( w \neq 0 \) and \( y \neq 0 \), then \( wy \neq 0 \). Since \( xz \) equals \( wy \) times an integer and \( wy \neq 0 \), then \( wy \) divides \( xz \). ∎
*There is something wrong with the proof, so please do not tell me that the proof is actually correct. Thank you.*
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