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Ierofanga [76]
3 years ago
10

This table gives a few (x,y)(x,y)left parenthesis, x, comma, y, right parenthesis pairs of a line in the coordinate plane. -28 -

54 -21 -40 -14 -26 What is the y-intercept of the line?
Mathematics
1 answer:
Serhud [2]3 years ago
7 0

Answer:

The y-intercept is y = 2

Step-by-step explanation:

A linear function has the following format:

y = mx + b

In which m is the slope and b is the y-intercept, which is the value of y when x = 0.

We are given these three points:

(-28, -54)

(-21, -40)

(-14, -26)

We use two of them to build a system, to find values for m and b.

(-28, -54)

This means that when x = -28, y = -54

So

y = mx + b

-54 = -28m + b

28m = b + 54

m = \frac{b + 54}{28}

(-21, -40)

This means that when x = -21, y = -40

So

y = mx + b

-40 = -21m + b

-40 = -21\frac{b + 54}{28} + b

-40 = \frac{-21b - 1134 + 28b}{28}

7b - 1134 = -40*28

7b = 14

b = \frac{14}{7}

b = 2

The y-intercept is y = 2

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Use euler's theorem to find a number x between 0 and 28 with x85 congruent to 6 modulo 35.
Elan Coil [88]
Problem: find 0 ≤ x ≤ 28 such that x^85 ≡ 6 modulo 35.

By Fermat-Euler theorem:
If a and n are coprime, i.e. (a,n)=1, then
a^phi(n) ≡ 1 mod n  
where phi(n)=totient function, the number of positive integers less than n that is coprime with n.

for n=35, phi(35)=24 calculated as follows:
There are 10 positive integers from 1 to 34 which are NOT coprime with 35, namely {5,7,10,14,15,20,21,25,28,30}.Therefore phi(35)=34-10=24

From Fermat-Euler theorem, 
x^(phi(35) = x^24 ≡ 1 modulo 35  since (24,35)=1, i.e. 24 and 35 are coprime.
=>
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and 

x^85 ≡ x^(85-3*24) ≡ x^(85-72) ≡ x^(13) ≡ 6 mod 35 ............(2)

Substituting (1) in (2)
x^(12)*x ≡ 6 mod 35
=>
(+1)*x = 6 mod 35   or  (-1)*x ≡ 6 mod 35
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=>  x=6 

So

6^85 ≡ 6 mod 35


If any clarifications are needed or if you find any errors, please post.
5 0
3 years ago
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