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kaheart [24]
3 years ago
14

If 2^n + 1 is an odd prime for some integer n, prove that n is a power of 2. (H

Mathematics
1 answer:
vovikov84 [41]3 years ago
4 0

Step-by-step explanation:

We will prove by contradiction. Assume that 2^n + 1 is an odd prime but n is not a power of 2. Then, there exists an odd prime number p such that p\mid n. Then, for some integer k\geq 1,

n=p\times k.

Therefore

  1. 2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p.

Here we will use the formula for the sum of odd powers, which states that, for a,b\in \mathbb{R} and an odd positive number n,

a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-...+b^{n-1})

Applying this formula in 1) we obtain that

2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p=(2^k+1)(2^{k(p-1)}-2^{k(p-2)}+...-2^{k}+1).

Then, as 2^k+1>1 we have that 2^n+1 is not a prime number, which is a contradiction.

In conclusion, if 2^n+1 is an odd prime, then n must be a power of 2.

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vodomira [7]
X = 1/4, 5, -8

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3 0
2 years ago
Read 2 more answers
If Y varies directly as X and Y= 79 when X= 49, Find Y if X = 9
ryzh [129]

Answer:

y = 14.5

Step-by-step explanation:

given that y varies directly as x

mathematically,

y ∝ x

y = kx .............................................. where k is the constant of proportionality.

k = y /x

given that

y = 79

x = 49

k = 79/49

k = 1.61

to find y when x = 9

from

y = kx

y = 1.61 × 9

y = 14.5

therefore the value of y when x = 9 in ths variation is evaluated to be equals to 14.5

7 0
3 years ago
The intensity, or loudness, of a sound can be measured in decibels (dB), according to the equation I (d B) = 10 log left-bracket
laila [671]

Answer:

<h2>The intensity in decibel is 320 decibel </h2>

Step-by-step explanation:

Given the intensity, or loudness, of a sound  measured in decibels (dB), according to the equation I (dB)= 10log(\frac{I}{Io} ) where;

I is the intensity of a given sound and

Io is the threshold of hearing intensity

To get I(dB) when I=10^{32} Io

We will substitute the value of I = I=10^{32} Io into the equation above to have;

I (dB)= 10log(\frac{10^{32}Io }{Io} )\\I(dB)=10log10^{32}\\ I(dB)=32*10log10\\

Since log10 = 1;

I(dB)=32*10(1)\\I(dB)=320

The intensity in decibel is 320 decibel

7 0
4 years ago
Read 2 more answers
5. As two boats are coming to the end of a race, a top view of their positions is
Dafna1 [17]

The distance of the boat on the right to the finish line is 297.29 m and the distance of the boat on the left to the finish is 294.11 m, so the boat on the left will reach the finish line faster.

<h3>How do you know which boat will reach the finish line faster? </h3>

The image shows two triangles of which we know an angle and the value of the opposite leg. Therefore, we can perform a mathematical operation to know the value of the hypotenuse of these triangles, which would be the distance of the boats to the goal as shown below.

h = hypotenuse

  • Right Boat

Sen 22° =  \frac{110 m}{h}

h = \frac{110 m}{ Sin 22}

h = \frac{110 m}{0.37}

h = 297,29 m

  • Left boat

Sen 20° = \frac{100}{h}

h = \frac{|00 m}{ Sin 20}

h = \frac{100 m}{0.34}

h = 294,11 m

According to the above, the distance of the boat on the left is less, so it can be inferred that this boat will reach the finish line faster.

Learn more about triangles in: brainly.com/question/2269348

8 0
2 years ago
If x=7+root 40 ,find the value of root x+1by the root of x​
Luda [366]

Chapter : Algebra

Study : Math in Junior high school

x = 7 + √40

find √x of √x + 1

= √x + 1

= √(7+√40) + 1

in Formula is :

= √7+√40 = √x + √y

= (√7+√40)² = (√x + √y)²

= 7+√40 = x + 2√xy + y

= 7 + √40 = x + y + 2√xy

→ 7 = x + y → y = 7 - x ... Equation 1

→ √40 = 2√xy → √40 = 2.2√10 = 4√10

= xy = 10 ... Equation 2

substitution Equation 1 to 2 :

= xy = 10

= x(7-x) = 10

= 7x - x² = 10

= x² - 7x + 10 = 0

= (x - 5)(x - 2) = 0

= x = 5 or x = 2

Subsitution x = 5 and x = 2, to equation 1

#For x = 5

= y = 7 - x

= y = 7 - (5)

= y = 2

#For x = 2

= y = 7 - x

= y = 7 - (2)

= y = 5

and his x and y was find :

#Equation 1 :

= x = 5 and y = 2

#Equation 2 :

= x = 2 and y = 5

So that :

√7+√40 = √x + √y

= √7+√40 = √2 + √5

And that is answer of question :

= √2 + √5 + 1

3 0
3 years ago
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