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AleksandrR [38]
3 years ago
7

Let n be a natural number. Show that 3 | n3 −n

Mathematics
1 answer:
Vladimir [108]3 years ago
4 0
Let n=1. Then n^3-n=1^3-1=0. By convention, every non-zero integer n divides 0, so 3\vert n^3-n.

Suppose this relation holds for n=k, i.e. 3\vert k^3-k. We then hope to show it must also hold for n=k+1.

You have

(k+1)^3-(k+1)=(k^3+3k^2+3k+1)-(k+1)=(k^3-k)+3(k^2+k)

We assumed that 3\vert k^3-k, and it's clear that 3\vert 3(k^2+k) because 3(k^2+k) is a multiple of 3. This means the remainder upon divides (k+1)^3-(k+1) must be 0, and therefore the relation holds for n=k+1. This proves the statement.
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The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long.
Otrada [13]

Answer:

The dock is 4.58 ft high

Step-by-step explanation:

we know that

Applying the Pythagoras theorem

Let

x------> the height of the dock

11^2=x^2+10^2

Solve for x

x^2=11^2-10^2

x^2=121-100

x^2=21

x=\sqrt{21} ft

x=4.58 ft

5 0
2 years ago
Carlos biked 24 mi and ran 6 mi.
Leviafan [203]
1:4
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6 0
3 years ago
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please please help me I have a test I need to take! Answer the following questions about the centroid of this triangle!!!!
maria [59]

Answer: I am sorta answering this on my alt, just move along

Step-by-step explanation:

The whole lenght is the short length 9 and the long length 18 added.

3 0
3 years ago
****BRAINLIEST GOES TO THE FIRST CORRECT ANSWER****
Drupady [299]

The two equations are vertical angles, which mean they are equal.

Set each equation to equal each other and solve for x.


3x-3 = 6(x-10)

Simplify the right side:

3x-3 = 6x-60

Subtract 3x from each side:

-3 = 3x - 60

Add 60 to each side:

57 = 3x

Divide both sides by 3:

x = 57 / 3

x = 19



5 0
3 years ago
When adding or subtracting with scientific notation,why is it important to have the same power of 10?
sladkih [1.3K]
<h2>Explanation:</h2><h2></h2>

Because we perform something like common factor, so this guarantee that corresponding digits have regarding coefficients have the same place value. Remember that a number written in scientific notation is given by the product of a number between 1 and 10 and another number that is power of 10. For example, let's write two examples about this:

Addition:

a=2\times 10^3 \\ \\ b=4\times 10^3 \\ \\ \\ Then: \\ \\ a+b=2\times 10^3+4\times 10^3 \\ \\ =(2+4)\times 10^3 \\ \\ =5\times 10^3

Addition:

a=2\times 10^3 \\ \\ b=4\times 10^3 \\ \\ \\ Then: \\ \\ a+b=2\times 10^3+4\times 10^3 \\ \\ =(2+4)\times 10^3 \\ \\ =6\times 10^3

Subtraction:

a=7\times 10^5 \\ \\ b=2\times 10^5 \\ \\ \\ Then: \\ \\ a-b=7\times 10^5-2\times 10^5 \\ \\ =(7-2)\times 10^5 \\ \\ =5\times 10^5

4 0
3 years ago
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