7. Let a, b, c € Z be integers with a and b not both 0. Let d = gcd(a, b). (a) Prove that there exist x, y € Z such that if and only if d divides c. (b) Suppose there exist xo, yo € Z such that axo + byo Show that for every k EZ, the numbers x = xo + ax + by = C kb d are integers and ax + by = c. (c) Suppose still that xo, yo € Z satisfy x = xo + and = C. and axo+byo Show that if x, y EZ satisfies the equation ax + by = c, then ka kb d d y = yo = C. y = yo ka d for some k E Z. (d) Use the results from parts (a)-(c) to explain why the equation = 18x + 42y has integer solutions, and find all integer solutions x, y € Z. €30
7. Let a, b, c € Z be integers with a and b not both 0. Let d = gcd(a, b). (a) Prove that there exist x, y € Z such that if and only if d divides c. (b) Suppose there exist xo, yo € Z such that axo + byo Show that for every k EZ, the numbers x = xo + ax + by = C kb d are integers and ax + by = c. (c) Suppose still that xo, yo € Z satisfy x = xo + and = C. and axo+byo Show that if x, y EZ satisfies the equation ax + by = c, then ka kb d d y = yo = C. y = yo ka d for some k E Z. (d) Use the results from parts (a)-(c) to explain why the equation = 18x + 42y has integer solutions, and find all integer solutions x, y € Z. €30
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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![7. Let \( a, b, c \in \mathbb{Z} \) be integers with \( a \) and \( b \) not both 0. Let \( d = \gcd(a, b) \).
(a) Prove that there exist \( x, y \in \mathbb{Z} \) such that
\[ ax + by = c \]
if and only if \( d \) divides \( c \).
(b) Suppose there exist \( x_0, y_0 \in \mathbb{Z} \) such that
\[ ax_0 + by_0 = c. \]
Show that for every \( k \in \mathbb{Z} \), the numbers
\[ x = x_0 + \frac{kb}{d} \quad \text{and} \quad y = y_0 - \frac{ka}{d} \]
are integers and \( ax + by = c \).
(c) Suppose still that \( x_0, y_0 \in \mathbb{Z} \) satisfy
\[ ax_0 + by_0 = c. \]
Show that if \( x, y \in \mathbb{Z} \) satisfies the equation \( ax + by = c \), then
\[ x = x_0 + \frac{kb}{d} \quad \text{and} \quad y = y_0 - \frac{ka}{d} \]
for some \( k \in \mathbb{Z} \).
(d) Use the results from parts (a)–(c) to explain why the equation
\[ 18x + 42y = 30 \]
has integer solutions, and find all integer solutions \( x, y \in \mathbb{Z} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7626d59-6405-4948-a625-19aca32c9eb7%2Fb1a49e4b-aa00-4fc2-ad0e-cd56db70e667%2F3wisdjd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. Let \( a, b, c \in \mathbb{Z} \) be integers with \( a \) and \( b \) not both 0. Let \( d = \gcd(a, b) \).
(a) Prove that there exist \( x, y \in \mathbb{Z} \) such that
\[ ax + by = c \]
if and only if \( d \) divides \( c \).
(b) Suppose there exist \( x_0, y_0 \in \mathbb{Z} \) such that
\[ ax_0 + by_0 = c. \]
Show that for every \( k \in \mathbb{Z} \), the numbers
\[ x = x_0 + \frac{kb}{d} \quad \text{and} \quad y = y_0 - \frac{ka}{d} \]
are integers and \( ax + by = c \).
(c) Suppose still that \( x_0, y_0 \in \mathbb{Z} \) satisfy
\[ ax_0 + by_0 = c. \]
Show that if \( x, y \in \mathbb{Z} \) satisfies the equation \( ax + by = c \), then
\[ x = x_0 + \frac{kb}{d} \quad \text{and} \quad y = y_0 - \frac{ka}{d} \]
for some \( k \in \mathbb{Z} \).
(d) Use the results from parts (a)–(c) to explain why the equation
\[ 18x + 42y = 30 \]
has integer solutions, and find all integer solutions \( x, y \in \mathbb{Z} \).
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