[ equations in integers ]

Find all pairs of natural numbers that satisfy the equation $19 m+84 n=1984$.

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Let's write the equation as $19(m-100)=84(n-1)$. Since 19 is a prime number and 84 is not divisible by 19, then $n-1$ must be divisible by 19, that is, $n=19 k+1$, where

$k \geq 0$. Then $m=100-84 k$. Since $m$ is a natural number, $k=0$ or $k=1$.

## Answer

$(100,1),(16,20)$.