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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,638

Sorry, I have never heard of that notation.

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**zetafunc.****Guest**

ShivamS wrote:

Sorry, I have never heard of that notation.

[a] is also used, if the modulus has been given.

Hi zetafunc.,

How did you compute the modular inverse of 9?

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**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 120

We have a linear Diophantine equation.

9y - 17 = 11x

y-(17/9) = x + (2x/9)

y-x-1 = (8+2x)/9

Because x and y are integers so is (8+2x)/9.

(8+2x)/9=z

8+2x = 9z

4+x=4z+(z/2)

4+x-4z=(z/2)

z=2w

we have

8+2x=18w

so we have

x=9w - 4

y=11w - 3

So we have 2 solutions x=-4 and y=-3 also x=5 and y=8

To get the other number we subtract y from 11.

so we have 83 as one solution and -38 as another solution also -83 is another solution.

*Last edited by Fruityloop (2014-04-08 16:58:22)*

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x=9w - 4

y=11w - 3

How do you solve that?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**zetafunc.****Guest**

Agnishom wrote:

Hi zetafunc.,

How did you compute the modular inverse of 9?

What we want to do is find an a such that

.From here you could use trial and error and try all possible values of a until you find one that satisfies the above relation. This works well for a smaller modulo, but for larger ones, we need a better technique. A more efficient way of finding the inverse for larger numbers is to use the Euclidean algorithm.

If

, then p does not divide a, and hence a and p are coprime. By the h,k-lemma, there exist integers h,k such that ph + ak = 1. Then, in , and hence . We can find h,k via the Euclidean algorithm. In our example, we wish to compute gcd(9, 11). (This is trivial but we're not interested in the final result.)11 = (1 × **9**) + **2****9** = (4 × **2**) + **1**

Re-arranging, we see that 1 = (5 × 9) + (-4 × 11), and the modular inverse of 9 is 5.

**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,638

I have heard different countries use different notation. For example, here in the US, when considering an n-dimensional space, mathematicians write it as R^n,. At Cambridge, they write it as d-dimensional space denoted with R^d.

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