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denis-greek [22]
2 years ago
9

The mystery number has:

Mathematics
1 answer:
harkovskaia [24]2 years ago
8 0
The mystery number will be 19.4596
You might be interested in
Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64
tino4ka555 [31]

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}}

5 0
3 years ago
The bakery made 92 muffins. seventeen were blueberry 23 were cranberry and the rest were chocolate chip. how many chocolate chip
Scorpion4ik [409]
53 i think i could be wrong
6 0
3 years ago
Please someone help, give the right answer it’s important
KonstantinChe [14]

Step-by-step explanation:

Please see the attached picture and I hope I have given the right answer.

5 0
3 years ago
If 3
Sliva [168]

Answer:Given that 3 persons can weave 168 shawls in 14 days. Then, Shawls are woven in 1 day by the 3 men = 168/14 = 12. Shawls are woven in 1 day ...

Step-by-step explanation:

4 0
2 years ago
how can you use distributive property to write an expression equivalent to the one given above? Use reasoning to explain how you
Anit [1.1K]

The reason how the expression are equivalent given below:

<h3>What is Distributive Property?</h3>

Multiplying a number by a sum or difference is the same as multiplying by each number in the sum or difference and then adding or subtracting.

The Complete question is:

How can you use distributive property to write an expression equivalent to the one given above? Use reasoning to explain how you know the expressions are equivalent.

Distributive property represents distribute the outside number among the number or digits in parenthesis.

For Example: Solve the expression 7(20 + 3) using the distributive property of multiplication over addition.

When we solve the expression 7(20 + 3) using the distributive property,

we first multiply every addend by 7.

This is known as distributing the number 7 amongst the two addends and then we can add the products.

This means that the multiplication of 7(20) and 7(3) will be performed before the addition.

Hence,  7(20) + 7(3) = 140 + 21 = 161.

Learn more Distributive Property here:

brainly.com/question/5637942

#SPJ1

8 0
2 years ago
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