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# I don't understand this

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what is the modular inverse of 11 , modulo 1000?

Express your answer as an integer from 0 to 999 , inclusive.

Jul 10, 2020

#1
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what is the modular inverse of 11 , modulo 1000?

[91 x 11] mod 1000 = 1, which is the definition of "modular multiplicative inverse"

Jul 10, 2020
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+111628
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Melody  Jul 10, 2020
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int[1000 / 11] = 91. Hence, [11 x 91] mod 1000 = 1

Guest Jul 10, 2020
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+8352
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Let $$x \equiv 11^{-1}\pmod{1000}$$

Now, because $$11^{\varphi(1000)} \equiv 1 \pmod{1000}$$, where $$\varphi(x)$$ is the Euler totient function,

$$x\equiv 11^{\varphi(1000) - 1} \pmod{1000}$$

We proceed to find the value of $$\varphi(1000)$$.

$$\varphi(1000) = 1000\left(1 - \dfrac12\right)\left(1 - \dfrac15\right) = 400$$

Therefore

$$x\equiv 11^{399} \pmod{1000}$$

$$x\equiv 1331^{133} \pmod{1000}\\ x\equiv 331^{133}\pmod{1000}$$

Now,

$$331^{133}\equiv 3 \pmod 8$$

$$331^{133}\equiv 81^{33}\equiv3^{32} = 7^6 \cdot 9 \equiv 9216 \equiv 91\pmod{125}$$

By CRT, $$x \equiv 331^{133}\equiv 91 \pmod{1000}$$

This means $$11^{-1} \equiv 91 \pmod{1000}$$.

Jul 11, 2020
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Thank Max, another one for me to ponder :)

Melody  Jul 11, 2020