Question
When a number is divided by 30, 45 and 54, there is always a remainder of 29. Find the least possible number.
When a number is divided by 30, 45 and 54, there is always a remainder of 29. Find the least possible number.
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Koala Tutor
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Let $N$ be the number we seek. From the given information, we can write \begin{align*}
N &\equiv 29\pmod{30},\\
N &\equiv 29\pmod{45},\\
N &\equiv 29\pmod{54}.
\end{align*}From the first two congruences, we have \[N-29\equiv 0\pmod{30}\quad \text{and}\quad N-29\equiv 0\pmod{45}.\]Since 30 and 45 share no common divisors greater than 1, we have \[N-29\equiv 0\pmod{\operatorname{lcm}{(30,45)}}.\]From the third congruence, we have \[N-29\equiv 0\pmod{54}.\]Note that $\operatorname{lcm}{(30,45)}=90$, so \[N-29\equiv 0\pmod{90}.\]Thus, \[N\equiv 29\pmod{90}.\]The least number satisfying this congruence is $\boxed{29}$.
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