The non-zero natural number n is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223- \frac{3}{2} \cdot n$. Find the quotient of the division.
Problem
Source: Romania National Olympiad 2023
Tags: number theory, Divisibility
Gloona
23.04.2023 14:58
$$2023 = 14 \cdot144 +223 - \frac{3}{2} \cdot 144$$ This can easily be checked by writing the equation as $$2023=n \cdot q +223-\frac{3}{2}\cdot n$$ Then utilise the fact that n is a square.and perform some algebraic manipulations. You will get the above divison.
combo_nt_lover
10.07.2023 20:01
We have
\begin{align*}
2023 &= qn + \left(223- \frac{3}{2} \cdot n\right) \\
\iff 1800 &= \left(q - \frac{3}{2}\right)n \\
\iff 3600 &= (2q-3)n.
\end{align*}Since $3600$ is a perfect square and $n$ is also a perfect square, therefore we must have
$2q-3$ is a perfect square. Note that $3600=2^4\cdot 3^2\cdot 5^2$, therefore
$2q-3\in\{2^2, 2^4, 2^2\cdot 3^2, 2^2\cdot 5^2, 2^4\cdot 3^2, 2^4\cdot 5^2,
2^2\cdot 3^2\cdot 5^2, 2^4\cdot 3^2\cdot 5^2, 3^2, 5^2, 3^2\cdot 5^2\}$.
Note that
$2q-3$ is an odd number.
$223- \frac{3}{2} \cdot n$ is the remainder of the division, hence $0\le 223- \frac{3}{2} \cdot n\le n$.
The left inequality is equivalent to
$\frac{3}{2} \cdot n\le 223$, or $n\le 148$. The right inequality is
equivalent to $223\le\frac{5}{2}n$, or $n\ge 89$. These inequality leads to
$\left\lceil\frac{3600}{148}\right\rceil \le 2q-3 \le
\left\lfloor\frac{3600}{89}\right\rfloor$, or $25\le 2q-3\le 40$.
From these two observations, we have $2q-3=5^2$, from which we have $n=144$, $q=14$.
Dumb_at_math_1729
01.01.2024 21:16
After you find the limit of n,,,, Then you can say n should be 100 or 144 as it is perfect square. As we know 2q-3 is odd so 100×36=3600 If N=100 Then 2q-3 is even that is impossible. So it is 144