问题

I'm stuck on this cryptography problem using multiplication of a whole number and a fraction mod 10.

Here is the equation:

7 * (4/11) mod 10 =?

I know I am supposed to convert this to an integer since the mod operator does not work with fractions, but I cannot figure this one out. Obviously,

7 * (4/11) = 28/11,

but I cannot get the mod 10 of a fraction. The instructor wants the exact answer, not a decimal. Any help would be greatly appreciated!

回答1:

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

回答2:

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.

回答3:

I can speculate that the notation is wrong, and that the whole expression is supposed to be evaluated in mod 10 at each intermediate stage. Since ( 11 mod 1 ) is 1, then answer is (7 * 4) mod 10 = 8.

Imagine a calculator with support only for the ones digit.

I'm not saying this is the right answer, I agree 28/11 is the right answer as given, but I am trying to get into the head of the professor. This is common in cryptography, where every calculation is performed mod 2 ^ 256 or so.

回答4:

This is how the original question probably should have been written, as this has a different meaning. When the (mod 10) is written at the end, it means that each term is evaluated with an implied mod 10 operation.

$\sqrt{foo}$

The problem is a bit weird, as the modulo value of 10 is not general purpose, because it is not prime. For example, the following can not be evaluated because 1/2 mod 10 is not defined, because 2 and 10 are not coprime.

$\sqrt{foo}$

回答5:

So, here is the correct answer from the instructor. I have no idea how he came up with this:

    7  4/11 mod 10 = ((7  4) mod 10)(11−1 mod 10) mod 10
    = (28 mod 10)(1 mod 10) mod 10
    = (8)(1) mod 10
    = 8 mod 10

回答6:

Using Python:

from fractions import Fraction
from math import fmod

print (fmod(Fraction(28, 11), 10))

The result will be 2.545454545454. So I guess 8 is wrong.

来源：https://stackoverflow.com/questions/32418693/modular-arithmetic-using-fractions

标签

modulo

Modular arithmetic using fractions

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