15. For all integers a, b, and c, if a b and a c then a(b+c).
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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![**14.** Fill in the blanks in the following proof that for all integers \( a \) and \( b \), if \( a \mid b \) then \( a \mid (-b) \).
**Proof:** Suppose \( a \) and \( b \) are any integers such that \( a \mid b \). By definition of divisibility, there exists an integer \( r \) such that \( b = ar \). By substitution,
\[ -b = -(ar) = a(-r). \]
Let \( t = -r \). Then \( t \) is an integer because \( t = (-1) \cdot r \), and both \(-1\) and \( r \) are integers. Thus, by substitution, \( -b = at \), where \( t \) is an integer, and so by definition of divisibility, \( a \mid (-b) \), as was to be shown.
**15.** For all integers \( a \), \( b \), and \( c \), if \( a \mid b \) and \( a \mid c \) then \( a \mid (b+c) \).
**16.** For all integers \( a \), \( b \), and \( c \), if \( a \mid b \) then \( a \mid c \) then \( a \mid (b-c) \).
**17.** For all integers \( a \), \( b \), \( c \), and \( d \), if \( a \mid c \) and \( b \mid d \) then \( ab \mid cd \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ec5dee8-4b81-4c1e-b6ca-023c2626018b%2Fab2ccd25-8136-4ae2-9b1a-c2e0386d1ce1%2Fp25re1n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**14.** Fill in the blanks in the following proof that for all integers \( a \) and \( b \), if \( a \mid b \) then \( a \mid (-b) \).
**Proof:** Suppose \( a \) and \( b \) are any integers such that \( a \mid b \). By definition of divisibility, there exists an integer \( r \) such that \( b = ar \). By substitution,
\[ -b = -(ar) = a(-r). \]
Let \( t = -r \). Then \( t \) is an integer because \( t = (-1) \cdot r \), and both \(-1\) and \( r \) are integers. Thus, by substitution, \( -b = at \), where \( t \) is an integer, and so by definition of divisibility, \( a \mid (-b) \), as was to be shown.
**15.** For all integers \( a \), \( b \), and \( c \), if \( a \mid b \) and \( a \mid c \) then \( a \mid (b+c) \).
**16.** For all integers \( a \), \( b \), and \( c \), if \( a \mid b \) then \( a \mid c \) then \( a \mid (b-c) \).
**17.** For all integers \( a \), \( b \), \( c \), and \( d \), if \( a \mid c \) and \( b \mid d \) then \( ab \mid cd \).
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