Good 'ol Algebra

Level 2

Let:

$x^{3}-y^{2}=z^{5}$ ,

$x+z=y$

Where $x$ , $y$ and $z$ are all positive integers less than 19. Find $x+y+z$ .

The answer is 20.

2 solutions

Finn Hulse
Feb 8, 2014

Here are the numbers: $x$ is 7, $y$ is 10, and $z$ is 3. It's pretty cool, but I don't know quite how to solve it. But the total is 20.

I did it like this: after substituting $y=x+3$ into the first equation and manipulating, we get

$x^3-x^2-6x=z^5-9$ .

Factorizing now gives $x(x-3)(x+2)=z^5-9$ .

Since we know that $z^5-9\geq 0$ , we may test for integers $x \geq 4$ .

At $x=7$ there is a match for $z=3$ and $y=10$ .

Martin Falk - 7 years, 4 months ago

Nice!

Finn Hulse - 7 years, 4 months ago

Its better than yours dude

ashutosh mahapatra - 7 years, 2 months ago

@Ashutosh Mahapatra – Yeah. :D

Finn Hulse - 7 years, 2 months ago

Where and how did you got $y = x + 3$ , and $x ^ { 3 } - x ^ { 2 } \cdots$ , etc. @Martin Falk .

. . - 3 months, 1 week ago

These values satisfy the equations but what is the approach to it?

Satyam Choudhary - 7 years, 4 months ago

I don't know... I was wondering if you guys could help with that. ;)

Finn Hulse - 7 years, 4 months ago

well, i will keep ring o find a good approach o his question.

Satyam Choudhary - 7 years, 4 months ago

. .
Mar 3, 2021

$x ^ { 3 } - y ^ { 2 } = z ^ { 5 } \rightarrow x + z = y \Rightarrow, \text { we get } x = 7, y = 10, z = 3. \text { Lets check the answer. } 7 ^ { 3 } - 10 ^ { 2 } = 3 ^ { 5 } \rightarrow 343 - 100 = 243 \text { is true, and } 7 + 3 = 10 \text { is also true. }$

$\text { So, the answer is } 7 + 10 + 3 = \boxed { 20 }$

. . - 3 months, 1 week ago

Good 'ol Algebra

The answer is 20.

2 solutions

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