1 answer

Prove relatively prime

Question:

suppose a,b,n ? Z, with n>0. Suppose that ab=1 (mod n). Prove that both a and b are relatively prime.

*Z is the integers symbol and a,b,n are all integers


Answers

This is incorrect
5*5=1(mod(12))
5 and 5 are obviously not relatively prime

the question must have been both a and b are relatively prime to n

this can be proved as follows: suppose (a,n) = g
ab=1(modn) means ab-1=mn for some integer m

ab-mn=1
g|a and g|n so that g|ab-mn meaning g|1
hence g=(a,n)=1

similarly (b,n)=1

.

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