Come On, Come On, Dragonite!

Relevant wiki: Linear Diophantine Equations

Let the number of Dratinis caught be $x$ and the number of Dratinis hatched be $y$ . Therefore, we have,

$3x+10y=25$

As we know that $x$ and $y$ are both non-negative integers. So,

$x=5 \ , \ y=1$

Therefore, total Dratinis with Pi Han $= \boxed{6}$ .

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Why $(x,y) = (5,1)$ is the only solution?

Considering the diophantine equation $3x+10y=25$ , we have $\gcd(3,10) = 1$ . So,

$1 = 3 \cdot 7 + 10 \cdot (-2)$

Then one solution is $x^* = 7 \cdot 25$ and $y^* = (-2) \cdot 25$ . And all other solutions are of form,

$\left( x^* + m \frac{10}{\gcd(3,10)}, ~ y^* - m \frac{3}{\gcd(3,10)} \right)$

or

$\left( 7 \cdot 25 + 10m, ~ -50 - 3m \right)$

for some integer $m$ .

As per the problem, we need to find the integers that satisfy $7 \cdot 25 + 10m \ge 0$ and $-50 - 3m \ge 0$ , so,

$\dfrac{-7 \cdot 5}{2} \le m \le \dfrac{-50}{3}$

The only integer value of $m$ satisfying the above inequality is $m = -17$ , which yields our solution as:

$(x,y) = \bigg( 7 \cdot 25 + 10 \cdot (-17), ~ -50 - 3 \cdot (-17) \bigg) = \boxed{\left(5,1\right)}$