MAT 255 I. Let p and q be distinet prime mumbers and let a be an integer not divisible by p and not divisible by g. Show that the congruence *=a (mod py) has a unique solution modulo pq whenever the integer k is relatively prime to (p-1)(4-1).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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MAT 255
I. Let p and q be distinet prime mumbers and let a be an integer not divisible by p and tot divisible by
1. Show that the congruence
=a (mod py)
has a unique solution modulo pg whenever the integer k is relatively prime to (p-1)(4- 1).
2. Give an example illustrating the result above.
3. Generalize the result in Problem I to the case of the congruence
(mod pipaPs-p)
where p.Pa, Pi are distinet primes. Prove your generalized result.
4. Use the result in Problem 3 and state conditions on n, a, and k under which the congruence
a (umod n)
is solvable.
Transcribed Image Text:MAT 255 I. Let p and q be distinet prime mumbers and let a be an integer not divisible by p and tot divisible by 1. Show that the congruence =a (mod py) has a unique solution modulo pg whenever the integer k is relatively prime to (p-1)(4- 1). 2. Give an example illustrating the result above. 3. Generalize the result in Problem I to the case of the congruence (mod pipaPs-p) where p.Pa, Pi are distinet primes. Prove your generalized result. 4. Use the result in Problem 3 and state conditions on n, a, and k under which the congruence a (umod n) is solvable.
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