Proposition 5.5.20. Given a modular equation ax = c (mod b), where a,b, c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b.

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Please do Question 5.5.21 and the hint is to use Proposition 5.5.16. Please show everything and explain each step. 

Proposition 5.5.16. Given the Diophantine equation an + bm = c, where
a, b, c are integers. Then the equation has integer solutions for n and m if
and only if c is a multiple of the gcd of a and b.
PROOF. Since this is an "if and only if" proof, we need to prove it both
ways. We'll do “only if" here, and leave the other way as an exercise.
Since we're doing the "only if" part, we assume that an + bm = c is
solvable. We'll represent the gcd of a and b by the letter d. Since gcd(a, b)
divides both a and b, we may write a
a', b'. By basic algebra, we have an + bm =
this back in the original Diophantine equation, we get:
da' and b = db' for some integers
d(a'n + b'm). If we substitute
d(a'n + b'm) =
It follows that c is a multiple of, d, which is the gcd of a and b.
Transcribed Image Text:Proposition 5.5.16. Given the Diophantine equation an + bm = c, where a, b, c are integers. Then the equation has integer solutions for n and m if and only if c is a multiple of the gcd of a and b. PROOF. Since this is an "if and only if" proof, we need to prove it both ways. We'll do “only if" here, and leave the other way as an exercise. Since we're doing the "only if" part, we assume that an + bm = c is solvable. We'll represent the gcd of a and b by the letter d. Since gcd(a, b) divides both a and b, we may write a a', b'. By basic algebra, we have an + bm = this back in the original Diophantine equation, we get: da' and b = db' for some integers d(a'n + b'm). If we substitute d(a'n + b'm) = It follows that c is a multiple of, d, which is the gcd of a and b.
Proposition 5.5.20. Given a modular equation ax = c (mod b), where
a, b, c are integers. Then the equation has an integer solution for x if and
only if c is an integer multiple of the greatest common divisor of a and b.
Exercise 5.5.21. Prove both the "if" and the "only if" parts of Proposi-
tion 5.5.20. (*Hint*)
Transcribed Image Text:Proposition 5.5.20. Given a modular equation ax = c (mod b), where a, b, c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b. Exercise 5.5.21. Prove both the "if" and the "only if" parts of Proposi- tion 5.5.20. (*Hint*)
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