I'm not sure how to categorize this question.. I assume its number theory?

Eigendorf

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I recently took a test for a math competition for fun and I wasn't quite sure how to solve this problem.

The test is long over but I still remember the question which is as follows:

If X^2 + Y^2 + Z^7 = 2017 and X, Y, AND X, Y, and Z are positive integers, THEN

What is the sum of the values of X, Y, and Z?

It was a multiple choice question and the answers were 59 - 63 I believe

a) 59 b) 60 c) 61 d) 62 e) 63


First I tested what possible values for Z could exist.

**I tried Z = 1 which is 1^7 = 1
**I tried Z = 2 which is 2^7 = 128
**I tried Z = 3 which is 3^7 = 2187

This means the value of Z is 1 or 2. Z can not be zero because zero is neither positive nor negative (I assume?)

I then tested the upper bound for an integer that could exist that would be less than 2017

I came up with:
**44^2 = 1936
**45^2 = 2025

So a maximum integer value of 44 is possible for X or Y and a Maximum value of 2 is possible for Z.


I'm not really sure where to go from here though.. I tried plugging in Z = 2 and Z = 1 into the equation then solving for Y. The zero for the equation doesn't graph out to an integer.

Ie y = sqrt(2016 - X^2) for Z = 1 or y = sqrt(1889 - X^2) for Z = 2.

Any suggestions or pointers would be appreciated. I'm interested in this problem just out of curiosity.
 
Hmmm..

I decided to approach this from the backend..

First I started with the maximum value of X or Y and subtracted that from 2017..

Ie.. I iterated

2017 - 44^2 - 128 ... Then I tried the square root of the result..

I continued with sqrt(2017 - 43^2 -128) all the way down to

sqrt(2017 - 40^2 - 128) which equals 40

Then knowing that Z = 2 and X = 40 I can solve for Y = 17.

So the answer would be A.

Is there a more direct algorithm or method of solving this other than just iterating through possible solutions?
 
First, it's a terrible idea to respond to your own post. This makes it look like it's already answered and is, therefore, far less likely to get any response.

Second, it is difficult to respond, not knowing where the question came from or who holds the copyright on the examination.

Third, seems like you have some reasonable thoughts and processes. Why do you doubt?
 
I apologize.

I probably should have just edited the original post but I literally was just crunching numbers for a while after I posted the suggestion and I guess I got "lucky" and struck on the solution.

I'm assuming there is another way to solve the problem that is more viable on more difficult problems. I was just looking for a suggestion on a good approach to this other than just iterating.

I don't know if questions are copyrightable but this came from the AMATYC 2017 test. Please delete this thread if I violated some copyright clause.
 
I don't understand tk's concerns. Perhaps, he doesn't allow web sites to store cookies on his device. Regarding copyrights, I see no issue discussing a generic exercise taken from an old exam.

Your analysis that Z must be 1 or 2 is great. One could go in multiple directions, from there. You successfully went one way and found the answer. Here's another way.

My approach uses the Discriminant that appears in the Quadratic Formula, but it still requires checking cases and possibilities. The Discriminant is the radicand: B^2-4AC.

Knowing that Z is either 1 or 2, we would consider each case separately. Let's look at the case where Z=2.

In this case, the given equation simplifies to:

X^2 + Y^2 = 1889

and the possible choices simplify to:

X + Y = 57
X + Y = 58
X + Y = 59
X + Y = 60
X + Y = 61

Considering the first choice, we can write:

Y^2 = 1889 - X^2

Y = 57 - X

Squaring this and substituting it for that gives us a quadratic equation:

(57 - X)^2 = 1889 - X^2

Putting this into standard form, we see the coefficients {A,B,C}:

2X^2 - 114X + 1360 = 0

Now, in order for X to be an Integer, the Discriminant must be the square of an Integer, yes?

(-114)^2 - 4(2)(1360) = 2016

2016 is the square of an Integer, so this choice works out. Hence, the answer must be 59.

This approach could be streamlined (helpful, as there are potentially 10 possibilities to check):

Is [2M - u^2] an Integer squared?

Here, M corresponds to the 1889 part and u corresponds to the 57 part.

Feel free to ask any questions about this. :cool:
 
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