Question
Download Solution PDFLet k be the order of a mod n then ab ≡ 1(mod n) if and only if
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Let b ∈ Z such that ab ≡ 1(mod n).
Let us apply division algorithm to b and k then we have,
b = kq + r where 0 ≤ r ≤ k
Consider, ab = akq + r = (ak)q . ar
By hypothesis ab ≡ 1(mod n) and ak ≡ 1(mod n).
Hence, ar ≡ 1(mod n) where 0 ≤ r ≤ k
∴ r has to be equal to zero and otherwise the choice of k has the smallest positive integer such that ak ≡ 1(mod n) will be contradicted.
Hence, b = qk
⇒ k | b
⇒ k divides b
Hence, the correct answer is option 2)
Last updated on Jun 13, 2025
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