Let $a\equiv (3^{-1}+5^{-1})^{-1}\pmod{11}$ . What is the remainder when a is divided by 11?
+118608 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.
