Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97/41x = 2 Modulo 97

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Example of Modulo Arithmetic

The solution of the integer congruence:

$41 x \equiv 2 \pmod {97}$

is:

$x = 45$


Proof

From Multiplicative Inverse of $41 \pmod {97}$ we have:

$41 \times 71 = 1 \pmod {97}$


Thus:

\(\ds 41 x\) \(\equiv\) \(\ds 2\) \(\ds \pmod {97}\)
\(\ds \leadsto \ \ \) \(\ds 71 \times 41 x\) \(\equiv\) \(\ds 71 \times 2\) \(\ds \pmod {97}\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\equiv\) \(\ds 142\) \(\ds \pmod {97}\)
\(\ds \) \(\equiv\) \(\ds 45\) \(\ds \pmod {97}\)


Hence the result.

$\blacksquare$


Sources