25. The interior angles of a triangle are $(5 x+3 y)^{\circ}$, $(3 x+20)^{\circ}$ and $(10 y+30)^{\circ}$, where $x, y$ are positive integers.
What is the value of $x+y$ ?
A 15
B 14
C 13
D 12
E 11
Solution: A
The angles of a triangle add up to $180^{\circ}$. So $(5 x+3 y)+(3 x+20)+(10 y+30)=180$. This gives $8 x+13 y+50=180$. Therefore $8 x+13 y=130$, and so
$$
8 x=130-13 y=13(10-y) .
$$

It follows that $8 x$ is a multiple of 13 and hence, as 8 and 13 have no common factors, it must be that $x$ is a multiple of 13 . So $x=13 a$, where $a$ is a positive integer. It then follows from (*) that
$$
8(13 a)=13(10-y)
$$
and so, cancelling the factor 13
$$
8 a=10-y,
$$
and hence $10-y$ is a positive number which is a multiple of 8 . The only positive value of $y$ for which $10-y>0$ and $10-y$ is a multiple of 8 , is $y=2$, which gives $a=1$ and hence $x=13$. So $x+y=13+2=15$.

Notes
In this problem we have just one equation, (*), with two unknowns. If there were no restrictions on the values of $x$ and $y$, this equation would have infinitely many solutions. However, here we are told that $x$ and $y$ are both positive integers. This additional information means that there is just one solution which we found above.

The term Diophantine equation is used for equations where we are only interested in positive integer solutions. The name comes from Diophantus of Alexandria who is believed to have lived around $250 \mathrm{AD}$, and whose influential book Arithmetica discussed problems of this type.
There are many well known Diophantine equations. For example, the fact that $\sqrt{2}$ is an irrational number is equivalent to the statement that the Diophantine equation $x^{2}-2 y^{2}=0$ has no solutions, as this equation may be rearranged as $\left(\frac{x}{y}\right)^{2}=2$.

Pythagorean triples are positive integer solutions of the Diophantine equation $x^{2}+y^{2}=z^{2}$. The French mathematician Pierre de Fermat (1601-1665) conjectured that none of the Diophantine equations $x^{n}+y^{n}=z^{n}$, where $n$ is a positive integer greater than 2 , has a solution where $x, y$, and $z$ are positive integers. It was not until 1994 that Andrew Wiles proved that Fermat's "Last Theorem" was correct. Wiles' proof uses some very sophisticated mathematics - the theory of elliptic curves and modular forms.