1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Ilya [14]
3 years ago
9

Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64

Mathematics
1 answer:
tino4ka555 [31]3 years ago
5 0

a.

x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}

By Fermat's little theorem, we have

x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5

x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7

5 and 7 are both prime, so \varphi(5)=4 and \varphi(7)=6. By Euler's theorem, we get

x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5

x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7

Now we can use the Chinese remainder theorem to solve for x. Start with

x=2\cdot7+5\cdot6

  • Taken mod 5, the second term vanishes and 14\equiv4\pmod5. Multiply by the inverse of 4 mod 5 (4), then by 2.

x=2\cdot7\cdot4\cdot2+5\cdot6

  • Taken mod 7, the first term vanishes and 30\equiv2\pmod7. Multiply by the inverse of 2 mod 7 (4), then by 6.

x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6

\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}

b.

x^5\equiv3\pmod{64}

We have \varphi(64)=32, so by Euler's theorem,

x^{32}\equiv1\pmod{64}

Now, raising both sides of the original congruence to the power of 6 gives

x^{30}\equiv3^6\equiv729\equiv25\pmod{64}

Then multiplying both sides by x^2 gives

x^{32}\equiv25x^2\equiv1\pmod{64}

so that x^2 is the inverse of 25 mod 64. To find this inverse, solve for y in 25y\equiv1\pmod{64}. Using the Euclidean algorithm, we have

64 = 2*25 + 14

25 = 1*14 + 11

14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

=> 1 = 9*64 - 23*25

so that (-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}.

So we know

25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}

Squaring both sides of this gives

x^4\equiv1681\equiv17\pmod{64}

and multiplying both sides by x tells us

x^5\equiv17x\equiv3\pmod{64}

Use the Euclidean algorithm to solve for x.

64 = 3*17 + 13

17 = 1*13 + 4

13 = 3*4 + 1

=> 1 = 4*64 - 15*17

so that (-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}, and so x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}

You might be interested in
12x−5y=−20 y=x+4 x=x=x, equals y=y=y, equals
aksik [14]

Answer:

The solution is (0, 4)

Step-by-step explanation:

Please pay attention to the first two equations and drop the last two:

12x−5y=−20 y=x+4 x=x=x, equals y=y=y   should ideally be:

12x−5y=−20

y=x+4

Let's find x.  Substitute x + 4 for y in the first equation, obtaining:

12x - 5(x + 4) = -20

Carrying out the indicated multiplication, we get:

12x - 5x - 20 = -20, or 7x = 0

If x = 0 then y must be 0 + 4, or 4.  

The solution is (0, 4)

8 0
3 years ago
Can someone please help me? Thank you!
Volgvan

Answer:

Step-by-step explanation:

1) 5^{\frac{2}{3} } is in exponential form.

<u>Now, radical form is </u>\sqrt[3]{5^{2} }

2) 5^{\frac{1}{2} } is in exponential form.

<u>Radical form is </u>\sqrt{5}

3) 3^{\frac{2}{5} } is in exponential form.

<u> Radical form is </u>\sqrt[5]{3^{2} }

4) 3^{\frac{5}{2} } is in exponential form.

<u> Radical form is </u>\sqrt{3^{5} }

<h3><u>If you need to ask any question, please let me know.</u></h3>
5 0
2 years ago
For 1-3 write an equivalent expression <br> 1.) -3(7+5g) <br> 2.) (x+7)+3y <br> 3.) 2/9-1/5*x
alukav5142 [94]

\huge \bf༆ Answer ༄

The equivalent expressions are ~

<h3>Question : 1 </h3>

  • \sf \:  - 3(7 + 5g)

  • \sf - 21 - 15g

  • \sf- 15g - 21
<h3>Question : 2</h3>

  • \sf \:( x + 7) + 3y

  • \sf \: x + 7 + 3y

  • \sf \: x + 3y + 7
<h3>Question : 3 </h3>

  • \sf \dfrac{2}{9}  -  \dfrac{1}{5x}

  • \sf \dfrac{10x  - 9}{45x}

7 0
2 years ago
Read 2 more answers
Who is your favorite super hero​
Stolb23 [73]

Superman is my favorite super hero

5 0
3 years ago
Read 2 more answers
Https://prnt.sc/15rb0bk really need help
Katena32 [7]

Answer:

x = 25

Step-by-step explanation:

Both angles are supplementary and will add up to 180°

5x - 5 + 2x + 10 = 180

collect like terms

5x + 2x +10 -5 = 180

7x + 5 = 180

7x = 180 - 5

7x = 175

x = 175/7

x=25

8 0
3 years ago
Other questions:
  • Three students earned $48.76 at the bake sell. The students split the earnings evenly, how much did each student recieve?
    11·2 answers
  • Can someone please help me with my math problem?
    12·1 answer
  • Determine a when d=6, g=4, k=3
    7·1 answer
  • Margo borrows $500, agreeing to pay it back with 3% annual interest after 14
    11·1 answer
  • Simplify -4+ -3 + 6
    6·1 answer
  • Natural selection acts by taking advantage of natural variations in the traits of organisms within a population. The ultimate so
    11·1 answer
  • John has 8 cm pieces of toy train track and Tom has 18 cm pieces of train track. How many of each
    11·2 answers
  • What are the possible degrees for the polynomial function?
    13·1 answer
  • the number in this sequence increase by 40 each time. 30 70 110 115... the sequence is continued with the same rule. Which numbe
    13·1 answer
  • HELP ME PLEASE I DONT KNOW HOW TO DO THIS
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!