Problem:
Factor $2289$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Factor the following integer}: \\
 2289 \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Since }\sqrt{2289}\approx 47.8, \text{it }\text{is }\text{sufficient }\text{to }\text{test }\text{2289 }\text{for }\text{divisibility }\text{by }\text{primes }\text{less }\text{than }\text{or }\text{equal }\text{to }47: \\
 \text{Test }\text{primes }\text{up }\text{to }\text{47 }\text{for }\text{factors }\text{of }2289 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{2289 }\text{is }\text{not }\text{divisible }\text{by }2 \text{since }\text{2289 }\text{is }\text{odd }\text{and }2 \text{is }\text{even}: \\
 2289=2289\text{(2289 }\text{is }\text{not }\text{divisible }\text{by }2) \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{l}
 \text{The }\text{sum }\text{of }\text{the }\text{digits }\text{of }\text{2289 }\text{is }2+2+8+9=21, \text{which }\text{is }\text{divisible }\text{by }3. \text{This }\text{means }\text{2289 }\text{is }\text{divisible }\text{by }3: \\
 2289=3\ 763: \\
\end{array}
 \\
 2289=3\times 763\text{(763 }\text{is }\text{not }\text{divisible }\text{by }2 \text{since }\text{2289 }\text{is }\text{not)} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{sum }\text{of }\text{the }\text{digits }\text{of }\text{763 }\text{is }7+6+3=16, \text{which }\text{is }\text{not }\text{divisible }\text{by }3. \text{This }\text{means }\text{763 }\text{is }\text{not }\text{divisible }\text{by }3: \\
 2289=3\times 763\text{(763 }\text{is }\text{not }\text{divisible }\text{by }2 \text{or }3) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{last }\text{digit }\text{of }\text{763 }\text{is }\text{not }5 \text{or }0, \text{which }\text{means }\text{763 }\text{is }\text{not }\text{divisible }\text{by }5: \\
 2289=3\times 763\text{(763 }\text{is }\text{not }\text{divisible }\text{by }2,3 \text{or }5) \\
\end{array}
 \\
 
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\begin{array}{l}
 \text{Divide }7 \text{into }763: \\
 
\begin{array}{ccccc}
 \text{} &  &  &  &  \\
  &  &  &  & \text{} \\
  &  &  &  & \text{} \\
 \text{} &  &  &  &  \\
 \text{} &  &  &  &  \\
 \text{} & \text{} &  &  &  \\
 \text{} &  &  &  & \text{} \\
 \text{} & \text{} & \text{} &  &  \\
\end{array}
 \\
\end{array}
 \\
 2289=3\times \fbox{$7\times 109$}\text{(109 }\text{is }\text{not }\text{divisible }\text{by }2,3 \text{or }5 \text{since }\text{763 }\text{is }\text{not)} \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{l}
 \text{Divide }7 \text{into }109: \\
 
\begin{array}{ccccc}
 \text{} & \text{} &  &  &  \\
  &  &  &  & \text{} \\
  &  &  &  & \text{} \\
 \text{} & \text{} &  &  &  \\
 \text{} &  &  &  & \text{} \\
 \text{} & \text{} & \text{} &  &  \\
\end{array}
 \\
 \text{109 }\text{is }\text{not }\text{divisible }\text{by }7: \\
\end{array}
 \\
 2289=3\times 7\times 109\text{(109 }\text{is }\text{not }\text{divisible }\text{by }2,3,5 \text{or }7) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{109 }\text{is }\text{not }\text{divisible }\text{by }\text{any }\text{prime }\text{less }\text{than }\sqrt{109}\approx 10.4 \text{(namely }2,3,5 \text{or }7). \text{Therefore }\text{109 }\text{is }\text{prime}: \\
 2289=3\times 7\times \fbox{$109$} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{There }\text{is }1 \text{copy }\text{of }3,1 \text{copy }\text{of }7 \text{and }1 \text{copy }\text{of }109 \text{in }\text{the }\text{product}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 2289=3\times 7\times 109 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/factor_integer_w_steps/2287.txt'}