Melody

Thanks EP

Firstly the expression you have written is not clear.

5^4x-3 

EP has interpretted this to mean   \(5^{4x-3}\)

Is this what you intended?  If so it should have been written as  5^(4x-3)

\(5^{4x-3}\\ =5^{4x}*5^{-3}\\ =(5^4)^x*\frac{1}{5^3}\\ =\frac{625^x}{125} \)

this is just another answer that is equivalent.

An aircraft flying with the wind flies to new york in 4 hours, on the return trip, the plane takes 5 hours in the same wind, find the speed of the plane if the air is still.

Let the distance be d km

With the wind the speed is d/4  km/hour

Against the wind the speed is d/5  km/hour

so

P+w=d/4

P-w=d/5

add

2P=9d/20

P=9d/40

So the speed of the plane with no wind would be     the distance in km * 9/40     km/hr

You want us to do all your homework for you?

Consider the sequence you discovered.

You found that  

\(1^1\;\mod2 = 1\;or \;-1\\ 2^2\;\mod3=+1\\ 3^3\;\mod4=-1\\ 4^4\;\mod5=+1\)

NOTE: If a number (mod n) = -1 Then the number divided by n will have a raimainder of  n-1

            If you want me to discuss this more I can.

So it looks like 

 \(n^n\mod(n+1)=+1 \text{ when n is even}\\ n^n\mod(n+1)=-1 \text{ when n is odd}\\\)

So can I prove that his is always the case??

\(n\mod(n+1)=-1\\ so\\ n^n\mod(n+1)\equiv (-1)^{n+1} \)

-1 raised to an even power will be +1     

-1 raised to an odd power will be -1

so

\(\frac{n^n}{n+1}\quad\text{has a remainder of +1 when n is an even positive integer}\\ and\\ \frac{n^n}{n+1}\quad\text{has a remainder of n when n is an odd positive integer}\\ \text{which means}\\ \frac{2019^{2019}}{2020}\quad\text{has a remainder of 2019 }\\\)

When you use capital letters on the internet it is interpreted as you screaming at people.

Which is quite impolite.

Plus why have you not thanked danne9000 or commented on the answer?

It seems you are the one who is failing to communicate.

If you google odd funtions and even functions then you will know what the terms mean and it will be easy.

Are you in the middle of a test?

I don't know.

when she started she had used none.

When she fisinches she will use 150

but since she is int the process of making it it could be any number in between.