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Number Theory

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Let  $a\equiv (3^{-1}+5^{-1})^{-1}\pmod{11}$ . What is the remainder when a is divided by 11?

Oct 19, 2021

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First I worked out what the inverse of 3 is mod 11

Let the inverse of 3 be x

3x=1 (mod11)

3x=11k+1

3x=9k+(2k+1)

x=3k+(2k+1)/3

Let b=(2k+1)/3

3b=2k+1

2k=2b+(b-1)

k=b+(b-1)/2

Let c=(b-1)/2

2c=b-1

b=2c+1

so   k=(2c+1)+c = 3c+1

and

x= 3(3c+1)+b = 9c+3+2c+1 = 11c+4 = 4 (mod11)

So the inverse of 3 mod 11 is 4 mod11

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Now I went through the same procedure and got that the inverse of 5 mod11 was -2 mod11

( I have not checked my working)

(4+-2)mod 11 = 2 mod 11

NowI went through the whole procedure again and found that the inverse of 2 mod 11 is  6 mod 11

So I got a is equivalent to 6 (mod11)

I have not checked my working and maybe there is an easier way to do it.

Oct 19, 2021