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\title{Divisibility I (Continues a Bit More)}
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\section*{Divisibility by 7 and 11.}


 \question{How can you quickly determine whether a number is divisible by 11? 
\newline \newline
Consider that the following numbers are divisible by eleven: \newline
  $121, 132, 143, 154, 165, 176, 187,  198, 209$ \newline
  $220, 231, 242, 253, 264, 275, 286, 297, 308$ \newline
  $319, 330, 341, 352, 363, 374, 385, 396, 407$ \newline
  $10021, 10032, 10043, 10054, 10065, 10076$ \newline
  $11022,11033, 11044, 11055, 11066, 11077$ \newline
  $12023, 12034, 12045, 12056, 12067, 12078$ \newline
  $12122, 12133, 12144, 12155, 12166, 12177$
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 \question{Now for the hard one, how can you reasonably quickly determine whether a number is divisible by 7? (The approach is related to the approach for 11.) 
 \newline \newline Consider that the following numbers are divisible by seven: \newline
 $77, 84, 91, 98$ \newline
 $105, 112, 119, 126, 133, 140, 147, 154, 161, 168$ \newline
 $203, 210, 217, 224, 231, 238, 245, 252, 259, 266$ \newline
 $301, 308, 315, 322, 329, 336, 343, 350, 357, 364$ \newline
 $931, 938, 945, 952, 959, 966, 973, 980, 987, 994$ \newline
 $1001, 1008, 1015, 1022, 1029, 1036, 1043, 1050$ \newline
 $2002, 2009, 2016, 2023, 2030, 2037, 2044, 2051$ \newline
 $3003, 3010, 3017, 3024, 3031, 3038, 3045, 3052$ \newline
 $10003, 10010, 10017, 10024, 11031$\newline
 $11004, 11011, 11018, 11025, 11032$\newline
 $20006, 20013, 20020, 20027, 20034$ \newline
 $100002, 100009, 100016, 100023$ \newline
 $999999, 1000006, 1000013$ 
 } 
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\section*{Making use of divisibility rules.}

\question{Figure out whether the following numbers are divisible by 2, 3, 5, 7 or 11. Circle the appropriate divisors. \newline
 \newline
 $85785: \quad \emph {2 \quad 3 \quad 5 \quad 7 \quad 11}$ \newline \newline
 $24794: \quad \emph {2 \quad 3 \quad 5 \quad 7 \quad 11}$ \newline \newline
 $80625: \quad \emph {2 \quad 3 \quad 5 \quad 7 \quad 11}$ \newline \newline
 $57233: \quad \emph {2 \quad 3 \quad 5 \quad 7 \quad 11}$ \newline \newline 
 $69286: \quad \emph {2 \quad 3 \quad 5 \quad 7 \quad 11}$
 } 

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\bigskip\question{List all the prime numbers less than 150. (A prime number is a positive integer that has only two divisors, 1 and itself. Note that one is not a prime number.) } 
  
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