The American Mathematics College is holding its orientation for incoming freshmen.
The incoming freshman class contains fewer than \(500\) people.
When the freshmen are told to line up in columns of \(23\), \(22\) people are in the last column.
When the freshmen are told to line up in columns of \(21\), \(14\) people are in the last column.
How many people are in the incoming freshman class?
\(\begin{array}{|rclrclr|} \hline x &\equiv& 22 \pmod{23} \\ \text{or } \quad x &=& 22+23n \\ x &\equiv& 14 \pmod{21} \\ \text{or}\quad \mathbf{ x } &=& \mathbf{14+21m} \\ \hline x = 14+21m &=& 22+23n \\ 21m &=& 23n + 8 \\ m &=& \dfrac{23n + 8}{21} \\ m &=& \dfrac{21n +2n + 8}{21} \\ m &=& n+ \underbrace{\dfrac{2n + 8}{21}}_{=a} \\ \mathbf{ m } &=& \mathbf{n+a} & a &=& \dfrac{2n + 8}{21} \\ & & & 21a &=& 2n + 8 \\ &&& 2n &=& 21a-8 \\ &&& n &=& \dfrac{21a-8}{2} \\ &&& n &=& \dfrac{20a+a-8}{2} \\ &&& n &=& 10a+\underbrace{\dfrac{a-8}{2}}_{=b} \\ &&& \mathbf{ n } &=& \mathbf{ 10a+b} & b=\dfrac{a-8}{2} \\ &&& & & & 2b= a-8 \\ &&& & & & \mathbf {a= 2b+8} \\ &&& n &=& 10(2b+8)+b \\ &&& \mathbf{n} &=& \mathbf{21b+80} \\ m &=& 21b+80+2b+8 \\ \mathbf{m} &=& \mathbf{23b+88} \\ \hline x &=& 14+21(23b+88) \\ x &=& 1862+483b \quad | \quad 1863 \mod{483} \equiv 413 \\ \mathbf{x} &=& \mathbf{413+483b} \\ \hline \end{array}\)
In the incoming freshman class are 413 people
