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# Modular Arithmetic

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Find 4^{-1}*9^{-1} (mod 35), as a residue modulo 35. (Give an answer between 0 and 34, inclusive.)

May 16, 2022

#1
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$$4^{-1} = \frac{1}{4}\\ 9^{-1} = \frac{1}{9}\\ \frac{1}{4} \cdot \frac{1}{9} = \frac{1}{36}$$

Therefore, the answer is 1/36 (mod 35).

May 16, 2022
#2
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A reasonable guess Pureant but unfortunately it is not correct.

* I will give you thumbs up for your attempt ;)

The multiplication inverse of  4 is 1/4  .

This is becasue  4 * 1/4 = 1

It is the same with modular arithemetic.

The number that is the inverse of 4 is the one that can be multipled by 4 to get 1.

So the inverse of 4 mod 35   is  B such that    4B=1  mod35

4*9=36 which is 1 mod 35

therefore the inverse of 4 is 9 and the inverse of 9 is 4  mod35

so the question becomes  9*4 = 36  = 1 mod35

May 16, 2022
edited by Melody  May 16, 2022
edited by Melody  May 16, 2022