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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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UNO [17]

Answer:

Inequality: x+10\leq 12

Inequality solved: x\leq 2

The graph is attached.

Step-by-step explanation:

You need to remember the meaning of the inequalities symbols:

: Less than.

> : Greater than.

\leq : Less than or equal to.

\geq : Greater than or equal to.

Let be "x" the number of more songs Carlotta can download.

According to the information given in the exercise, she can download no more than 12 song files per week. This indicates that you must use the symbol \leq.

Since she has already downloaded 10 song files this week, you can write the following inequality that represents this situation:

x+10\leq 12

In order to solve it, you must subtract 10 from both sides:

x+10-10\leq 12-10\\\\x\leq 2

Plot this result on a number line.

Since the symbol of the inequality is \leq, the dot must be filled (Observe the number line attached).

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3 years ago
100 aluminum cans weigh 5 pounds, If Ray collects 300 cans on clean up day, how many pounds of cants does he have?
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Find the inverse f(x) = 4x+2
alexdok [17]

Answer:

Let's solve for x.

y=4x+2

Step 1: Flip the equation.

4x+2=y

Step 2: Add -2 to both sides.

4x+2+−2=y+−2

4x=y−2

Step 3: Divide both sides by 4.

4x/4=y−2/4

x=1/4y+ −1/2

Answer:

x=1/4y+ −1/2

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Step-by-step explanation:

4 0
3 years ago
Use graph to answer question
slamgirl [31]

Answer:

cos(θ) = 3/5

Step-by-step explanation:

We can think of this situation as a triangle rectangle (you can see it in the image below).

Here, we have a triangle rectangle with an angle θ, such that the adjacent cathetus to θ is 3 units long, and the cathetus opposite to θ is 4 units long.

Here we want to find cos(θ).

You should remember:

cos(θ) = (adjacent cathetus)/(hypotenuse)

We already know that the adjacent cathetus is equal to 3.

And for the hypotenuse, we can use the Pythagorean's theorem, which says that the sum of the squares of the cathetus is equal to the square of the hypotenuse, this is:

3^2 + 4^2 = H^2

We can solve this for H, to get:

H = √( 3^2 + 4^2) = √(9 + 16) = √25 = 5

The hypotenuse is 5 units long.

Then we have:

cos(θ) = (adjacent cathetus)/(hypotenuse)

cos(θ) = 3/5

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3 years ago
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