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  2. 5
    • According to 5 sources
    5 is the modular inverse of 3 mod 7 since 5*3 mod 7 = 1
    Definition: An integer ā such that āa ≡ 1 (mod m) is said to be an inverse of a modulo m. Example: 5 is an inverse of 3 modulo 7 since 5∙3 = 15 ≡ 1 (mod 7)
    I need to find the inverse of 3 modulo 7. We have:. 3 x ≡ 1 (mod 7) This means:. 3 x = 1 + 7 a.for some integer a. Solve for x: x = 7 a + 1 3 ⇒ x = 2 a + a + 1 3 Since x is an integer, a + 1 must be a multiple of 3. The smallest case occurs when a = 2. Hence:. x = 2 (2) + 3 3 ⇒ x = 5 Therefore, 5 is the inverse of 3, modulo 7.
    The inverse of 3 (mod 7) is 5 since 3cdot5 = 15 and 15 (mod 7) equiv 1 (mod 7), therefore the first equation is equivalent to x equiv 25 (mod 7) equiv 4 (mod 7)...
    Similarly, 5 is a multiplicative inverse of 3 modulo 7.
     
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    One method is simply the extended Euclidean algorithm: \begin{align*} 31 &= 4(7) + 3\\\ 7 &= 2(3) + 1. \end{align*} So $ 1 = 7 - 2(3) = 7 - 2(31 - 4(7)) = 9(7) - 2(31)$. Viewing the equation $1 = 9(7) -2(31)$ modulo $31$ gives $ 1 \equiv …

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