Modular arithmetic using fractions

8

8 is the correct answer indeed.

7*4/11 mod 10 means we're looking at 7*4*x mod 10 where x is the modular inverse of 11 modulo 10, which means that 11*x mod 10 = 1. This is true for x=1 (11*1 mod 10 = 1)

So 7*4*x mod 10 becomes 7*4*1 mod 10 which is 28 mod 10 = 8

Have a look here: "Is it possible to do modulo of a fraction" on math.stackexchange.com.

One natural way to define the modular function is

a (mod b) = a − b ⌊a / b⌋

where ⌊⋅⌋ denotes the floor function. This is the approach used in the influential book Concrete Mathematics by Graham, Knuth, Patashnik.

This will give you 1/2(mod3)=1/2.

To work through your problem, you have a = 7 * (4/11) = 28/11, and b = 10.

a / b = (28/11)/10 = 0.25454545...

⌊a/b⌋ = 0

b ⌊a/b⌋ = 0 * 0 = 0

a - b ⌊a/b⌋ = 28/11 - 0 = 28/11

This means your answer is 28/11.

Wolfram Alpha agrees with me and gives 28/11 as the exact result. Google also agrees, but gives it as a decimal, 2.54545454.....

A fraction is an exact answer and not a decimal.