Problem:
Factor $228$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Factor the following integer}: \\
 228 \\
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 \\
\hline
 
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 \text{Since }\sqrt{228}\approx 15.0, \text{it }\text{is }\text{sufficient }\text{to }\text{test }\text{228 }\text{for }\text{divisibility }\text{by }\text{primes }\text{less }\text{than }\text{or }\text{equal }\text{to }15: \\
 \text{Test }\text{primes }\text{up }\text{to }\text{15 }\text{for }\text{factors }\text{of }228 \\
\end{array}
 \\
 
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\begin{array}{l}
 \text{The }\text{last }\text{digit }\text{of }\text{228 }\text{is }8, \text{which }\text{means }\text{it }\text{is }\text{even. }\text{Therefore }\text{228 }\text{is }\text{divisible }\text{by }2: \\
 228=2\ 114: \\
\end{array}
 \\
 228=2\times 114 \\
\end{array}
 \\
 
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\begin{array}{l}
 \text{The }\text{last }\text{digit }\text{of }\text{114 }\text{is }4, \text{which }\text{means }\text{it }\text{is }\text{even. }\text{Therefore }\text{114 }\text{is }\text{divisible }\text{by }2: \\
 114=2\ 57: \\
\end{array}
 \\
 228=2\times \fbox{$2\times 57$} \\
\end{array}
 \\
 
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 \text{57 }\text{is }\text{not }\text{divisible }\text{by }2 \text{since }\text{57 }\text{is }\text{odd }\text{and }2 \text{is }\text{even}: \\
 228=2\times 2\times 57\text{(57 }\text{is }\text{not }\text{divisible }\text{by }2) \\
\end{array}
 \\
 
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 \text{The }\text{sum }\text{of }\text{the }\text{digits }\text{of }\text{57 }\text{is }5+7=12, \text{which }\text{is }\text{divisible }\text{by }3. \text{This }\text{means }\text{57 }\text{is }\text{divisible }\text{by }3: \\
 57=3\ 19: \\
\end{array}
 \\
 228=2\times 2\times \fbox{$3\times 19$}\text{(19 }\text{is }\text{not }\text{divisible }\text{by }2 \text{since }\text{57 }\text{is }\text{not)} \\
\end{array}
 \\
 
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 \text{The }\text{sum }\text{of }\text{the }\text{digits }\text{of }\text{19 }\text{is }1+9=10, \text{which }\text{is }\text{not }\text{divisible }\text{by }3. \text{This }\text{means }\text{19 }\text{is }\text{not }\text{divisible }\text{by }3: \\
 228=2\times 2\times 3\times 19\text{(19 }\text{is }\text{not }\text{divisible }\text{by }2 \text{or }3) \\
\end{array}
 \\
 
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 \text{19 }\text{is }\text{not }\text{divisible }\text{by }\text{any }\text{prime }\text{less }\text{than }\sqrt{19}\approx 4.3 \text{(namely }2 \text{or }3). \text{Therefore }\text{19 }\text{is }\text{prime}: \\
 228=2\times 2\times 3\times \fbox{$19$} \\
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 \\
 
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 \text{There }\text{are }2 \text{copies }\text{of }2,1 \text{copy }\text{of }3 \text{and }1 \text{copy }\text{of }19 \text{in }\text{the }\text{product}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 228=2^2\times 3\times 19 \\
\end{array}
$} \\
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 \\
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{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/factor_integer_w_steps/226.txt'}