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Phoenix [80]
3 years ago
7

For each x and n, find the multiplicative inverse mod n of x. Your answer should be an integer s in the range 0 through n - 1. C

heck your solution by verifying that sx mod n = 1.(a) x = 52, n = 77(b) x = 77, n = 52(c) x = 53, n = 71(d) x = 71, n = 53
Mathematics
1 answer:
krok68 [10]3 years ago
6 0

Use the Euclidean algorithm to express 1 as a linear combination of x and n.

a. 52^{-1}\equiv40\pmod{77} because

77 = 1*52 + 25

52 = 2*25 + 2

25 = 12*2 + 1

so we can write

1 = 25 - 12*2 = 25*25 - 12*52 = (77 - 52)(77 - 52) - 12*52 = 77^2 - 2*52*77 + 52^2 - 12*52

Taken modulo 77 leaves us with

1\equiv52\cdot52-12\cdot52\equiv40\cdot52\pmod{77}\implies52^{-1}\equiv40\pmod{77}

b. First, 77\equiv25\pmod{52}, so really we're looking for the inverse of 25 mod 52. We've basically done the work in part (a) already:

1 = 25*25 - 12*52

Taken modulo 52, we're left with

1\equiv25\cdot25\pmod{52}\implies25^{-1}\equiv25\pmod{52}

c. The EA gives

71 = 1*53 + 18

53 = 2*18 + 17

18 = 1*17 + 1

so we get

1 = 18 - 17 = 3*18 - 53 = 3*71 - 4*53

so that taken module 71, we find

1\equiv(-4)\cdot53\pmod{71}\implies53^{-1}\equiv-4\equiv67\pmod{71}

d. Same process as with (b). First we have 71\equiv18\pmod{53}, and we've already shown that

1 = 3*18 - 53

which means, taken modulo 53, that

1\equiv3\cdot18\pmod{53}\implies71^{-1}\equiv18^{-1}\equiv3\pmod{53}

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Step-by-step explanation:

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Calculate the difference in the proportion of males and the proportion of females that smoke. Give your answer to 2 decimal plac
Xelga [282]

The difference in the proportion of males and the proportion of females that smoke is 0.08

<h3>Missing information </h3>

In a sample of 61 males, 15 smoke, while in a sample of 48 females, 8 smoke.

<h3>How to determine the proportion difference?</h3>

The given parameters are:

                             Male           Female

Sample                  61                  48

Smokers               15                   8

The proportion is calculated using:

p = Smoker/Sample

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Female = 8/48 = 0.17

The difference is then calculated as:

Difference = 0.25 - 0.17

Evaluate

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brainly.com/question/16828793

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5:45 ::  3: 3 + 2x + 10

You can reduce 5 / 45

5/45 = 1/9

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1:9 :: 3: 2x + 13

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2x + 13 - 13 = 27 - 13

2x = 14                              Divide by 2

2x/2 = 14/2                        Divide

x = 7

3 0
3 years ago
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