cmi-entrance 2020 QA10

cmi-entrance · India · ugmath Number Theory Modular Arithmetic Computation

Note that $25 \times 16 - 19 \times 21 = 1$. Using this or otherwise, find positive integers $a, b$ and $c$, all $\leq 475 = 25 \times 19$, such that

$a$ is $1 \bmod 19$ and $0 \bmod 25$,
$b$ is $0 \bmod 19$ and $1 \bmod 25$, and
$c$ is $4 \bmod 19$ and $10 \bmod 25$.

(Recall the mod notation: since 13 divided by 5 gives remainder 3, we say 13 is $3 \bmod 5$.)

Note that $25 \times 16 - 19 \times 21 = 1$. Using this or otherwise, find positive integers $a, b$ and $c$, all $\leq 475 = 25 \times 19$, such that
\begin{itemize}
  \item $a$ is $1 \bmod 19$ and $0 \bmod 25$,
  \item $b$ is $0 \bmod 19$ and $1 \bmod 25$, and
  \item $c$ is $4 \bmod 19$ and $10 \bmod 25$.
\end{itemize}
(Recall the mod notation: since 13 divided by 5 gives remainder 3, we say 13 is $3 \bmod 5$.)

Paper Questions

QA1 QA10 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB2 QB3 QB4 QB5 QB6