- #1

Werg22

- 1,427

- 1

a, b and c are positive integrers.

What is the sum of all possible values of a and b between 0 and 100 if

a^(-2) + b^(-2)=c^(-2)

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Werg22
- Start date

- #1

Werg22

- 1,427

- 1

a, b and c are positive integrers.

What is the sum of all possible values of a and b between 0 and 100 if

a^(-2) + b^(-2)=c^(-2)

- #2

gerben

- 510

- 1

[tex] a \in [0, 100] \rightarrow \sqrt{a} \in [0, 10] [/tex]

so the sum of all values of a is:

[tex] \sum_{x=1}^{10}{x^2}[/tex]

the same for b.

- #3

VietDao29

Homework Helper

- 1,426

- 3

I don't understand...gerben said:[tex]c = (\sqrt{a}+\sqrt{b})^2[/tex]

[tex]a ^ {-2} = \frac{1}{a ^ 2}[/tex],

Viet Dao,

Last edited:

- #4

Werg22

- 1,427

- 1

1/c^2=1/b^2 + 1/a^2

=(a^2+ b^2)/(ab)^2

c^2=ab^2/(a^2+ b^2)

c=ab/(a^2+ b^2)^1/2

Since a^2+ b^2 is an integrer, it's root is either another integrer or irrational. Thus in order for c to be an integrer, (a^2+ b^2)^1/2 must be an integrer.

Listing possible result;

3^2 + 4^2 = 5^2

5^2 + 12^2 = 13^2

20^2 + 21^2 = 29^2

...

Then considering the first possibility, a=4x and b=3x

c=12x^2/5x

=12x/5

We conclude that x must be a factor of 5. Since a=4x, a is multiple of 20. Adding up all multiples of 20 between 0 and 100

20 + 40 + 60 + 80 + 100=300

Now for b, b=3x

15 + 30 + 45 + 60 + 75 + 90=315

Now the second possibility,

c=60x/13

a=5x, b=12x

x must be a multiple of 13

5(13)=65, and 12(13)>100.

The next possibility, we knoe that a=20x and b=21x, and x has to be a factor of 29. Since 21(29)>20(29)>100, then there is no further solution.

300 + 315 + 65=680.

So the awnser is 680.

Share:

- Last Post

- Replies
- 7

- Views
- 413

- Last Post

- Replies
- 3

- Views
- 324

- Last Post

- Replies
- 3

- Views
- 564

- Last Post

- Replies
- 5

- Views
- 558

- Last Post

- Replies
- 3

- Views
- 551

- Last Post

- Replies
- 3

- Views
- 317

- Last Post

- Replies
- 4

- Views
- 471

- Last Post

- Replies
- 2

- Views
- 272

- Replies
- 11

- Views
- 639

- Last Post

- Replies
- 2

- Views
- 472