Let a, b, c € Z be integers with a and b not both 0. Let d = ged(a, b). (a) Prove that there exist r, y € Z such that if and only if d divides c. (b) Suppose there exist o, 30 € Z such that ax+by = c x = 10 + Show that for every k € Z, the numbers azo+byo = c. x = 10 + kb and d are integers and ar+by = c. (c) Suppose still that ro, yo € Z satisfy ka 6-d azo+byo = c. Show that if r, y € Z satisfies the equation ar+by=c, then kb ka and y=yo y=yod for some k € Z. (d) Use the results from parts (a)-(e) to explain why the equation 18x + 42y = 30 has integer solutions, and find all integer solutions z, y € Z.
Let a, b, c € Z be integers with a and b not both 0. Let d = ged(a, b). (a) Prove that there exist r, y € Z such that if and only if d divides c. (b) Suppose there exist o, 30 € Z such that ax+by = c x = 10 + Show that for every k € Z, the numbers azo+byo = c. x = 10 + kb and d are integers and ar+by = c. (c) Suppose still that ro, yo € Z satisfy ka 6-d azo+byo = c. Show that if r, y € Z satisfies the equation ar+by=c, then kb ka and y=yo y=yod for some k € Z. (d) Use the results from parts (a)-(e) to explain why the equation 18x + 42y = 30 has integer solutions, and find all integer solutions z, y € Z.
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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Transcribed Image Text: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 r, y € Z such that
if and only if d divides c.
(b) Suppose there exist o. yo Z such that
đạt
ax+by = c
azo +byo = c.
Show that for every ke Z, the numbers
kb
d
are integers and ax + by = c.
(c) Suppose still that ro, yo € Z satisfy
kb
x=x0+ d
and
y=yo-
azo+byo
Show that if x, y € Z satisfies the equation ax + by = C,
then
and
= C.
ka
d
y = yo
ka
d
for some k € 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 z, y € Z.
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