(2) Show that if a, b € Z/nZ has multiplicative orders k and I respectively and (k, l) = 1, then ab has multiplicative order kl. Hint: If (ab)r = 1 mod n then a” is the inverse of br modulo n. Use the previous exercise with m = -1 to deduce that the order of a" is the same as the order of br modulo n, and thus that r must be a multiple of kl.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 6E
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(2) Show that if a, b € Z/nZ has multiplicative orders k and I respectively and (k, l) = 1,
then ab has multiplicative order kl. Hint: If (ab) 1 mod n then a" is the inverse
of br modulo n. Use the previous exercise with m = -1 to deduce that the order of
a" is the same as the order of br modulo n, and thus that r must be a multiple of kl.
=
Transcribed Image Text:(2) Show that if a, b € Z/nZ has multiplicative orders k and I respectively and (k, l) = 1, then ab has multiplicative order kl. Hint: If (ab) 1 mod n then a" is the inverse of br modulo n. Use the previous exercise with m = -1 to deduce that the order of a" is the same as the order of br modulo n, and thus that r must be a multiple of kl. =
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