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3 years ago

8

If a system of equations has a solution What would the GraphAre look-alike

1 answer:

pickupchik [31]3 years ago

3 0

Answer:

their intersection point(s)

Step-by-step explanation:

The Solution to a system of equations is the point at which the lines intersect eachother

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Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$

Viktor [21]

When $n=0$ , we have

$5^{3^0} + 1 = 5^1 + 1 = 6$

$3^{0 + 1} = 3^1 = 3$

and of course 3 | 6. ("3 divides 6", in case the notation is unfamiliar.)

Suppose this is true for $n=k$ , that

$3^{k + 1} \mid 5^{3^k} + 1$

Now for $n=k+1$ , we have

$5^{3^{k+1}} + 1 = 5^{3^k \times 3} + 1 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k}\right)^3 + 1^3 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k} + 1\right) \left(\left(5^{3^k}\right)^2 - 5^{3^k} + 1\right)$

so we know the left side is at least divisible by $3^{k+1}$ by our assumption.

It remains to show that

$3 \mid \left(5^{3^k}\right)^2 - 5^{3^k} + 1$

which is easily done with Fermat's little theorem. It says

$a^p \equiv a \pmod p$

where $p$ is prime and $a$ is any integer. Then for any positive integer $x$ ,

$5^3 \equiv 5 \pmod 3 \implies (5^3)^x \equiv 5^x \pmod 3$

Furthermore,

$5^{3^k} \equiv 5^{3\times3^{k-1}} \equiv \left(5^{3^{k-1}}\right)^3 \equiv 5^{3^{k-1}} \pmod 3$

which goes all the way down to

$5^{3^k} \equiv 5 \pmod 3$

So, we find that

$\left(5^{3^k}\right)^2 - 5^{3^k} + 1 \equiv 5^2 - 5 + 1 \equiv 21 \equiv 0 \pmod3$

QED

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2 years ago

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Klio2033 [76]

Answer:78% countries

Step-by-step explanation:

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3 years ago

When you view the pendulum’s swing, it shows that at the very top of the swing KE = 0. What does that tell you about the pendulu

slava [35]

Answer:

The kinetic energy of an object is given by: Kinetic energy = 1/2 * mass * velocity If the kinetic energy of the pendulum at the top of its swing is 0, then this tells us that the velocity of the pendulum at that point is 0

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3 years ago

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Mashutka [201]

Answer:

its C

Step-by-step explanation:

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3 years ago

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2<br> 1 point<br> Find UV.<br> U V<br> 56<br> 45<br> --D<br> W

Tomtit [17]

Answer:

11

Step-by-step explanation:

$UV=UW-VW=56-45=11$

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2 years ago