## anonymous one year ago Explain why this problem is in a number theory book? I feel like it should be in a Cal 3 book. Prove that if (xₒ,yₒ) is a solution of the linear Diopahntine equation ax - by = 1, then the area of the triangle whose vertices are (0,0), (b,a) and (xₒ,yₒ) is 1/2? This could easily be proven using cross product

1. UsukiDoll

The Diophantine equation was in my number theory book as well. An equation which only integer solutions are allowed.

2. anonymous

eh, probably because it mentions linear Diophantine equations?

3. anonymous

And there is this problem too. A man pays \$1.43 for some apples and pears. If pears cost 17 cents each, and apples, 15 cents each, how many of each did he buy? O.O what the heck? this is 5th grade problem!

4. UsukiDoll

oh I had that problem too... hey are you reading Number Theory by George Andrews?

5. anonymous

@UsukiDoll yep :)

6. anonymous

anyways, this just follows from the determinant: $$\left|\det\begin{bmatrix}b&x_0\\a&y_0\end{bmatrix}\right|=\left|by_0-ax_0\right|=1$$ which is the area of the whose edges are given by $$(b,a),(x_0,y_0)$$, so the triangle is simply half that: $$1/2$$

7. UsukiDoll

I've used that book last year in my Elementary Number Theory class XDDDDDDDDDDD!!!!!!!! Luckily I didn't have to do that problem, but I was stunned when I saw that.

8. UsukiDoll

I've heard that it's the most popular book.

9. ganeshie8

Mostly the authors choose trivial examples in the start because the focus is on succcessfully applying a particular method, not on getting a quick answer

10. ganeshie8

the second question translates to solving linear diophantine eqn $17a+15p = 143$

11. anonymous

that problem is fine, it's a Diophantine equation, too: $$15a + 17p = 143\\ 15a=143\pmod{17}\implies -2a=-10\pmod{17}\implies a=5\pmod{17}\\17p=143\pmod{15}\implies 2p=10\pmod{17}\implies p=5\pmod{15}$$

12. anonymous

oops, the second line is broken

13. anonymous

$$2p=8\pmod{15}\implies p=8\pmod{15}$$

14. anonymous

O: I did the second problem by guess and check. Never mind about it being 5th a grade problem ^^

15. anonymous

I need someone to confirm this problem. Hold on...

16. anonymous