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|00000d40| 61 6e 20 69 6e 69 74 69 | 61 6c 20 76 65 6c 6f 63 |an initi|al veloc|
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|00000d60| 69 74 7b 6d 7d 2f 5c 75 | 6e 69 74 7b 73 7d 24 2c |it{m}/\u|nit{s}$,|
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|00000db0| 20 74 68 65 20 67 72 6f | 75 6e 64 2c 20 69 74 20 | the gro|und, it |
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|00000dd0| 74 20 24 38 30 74 2d 31 | 36 74 5e 7b 32 7d 5c 75 |t $80t-1|6t^{2}\u|
|00000de0| 6e 69 74 7b 6d 7d 24 2e | 20 46 69 6e 64 20 74 68 |nit{m}$.| Find th|
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|00000e20| 20 67 72 6f 75 6e 64 2e | 20 57 68 65 6e 20 64 6f | ground.| When do|
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|00000e40| 68 69 67 68 65 73 74 20 | 70 6f 69 6e 74 3f 5c 68 |highest |point?\h|
|00000e50| 66 69 6c 6c 20 5c 68 79 | 70 65 72 72 65 66 7b 25 |fill \hy|perref{%|
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|00000e80| 31 34 69 6e 7d 7b 30 2e | 32 30 35 69 6e 7d 7b 30 |14in}{0.|205in}{0|
|00000e90| 69 6e 7d 7d 7b 7d 7b 7d | 7b 52 6f 63 6b 65 74 7d |in}}{}{}|{Rocket}|
|00000ea0| 0d 0a 5c 65 6e 64 7b 65 | 6e 75 6d 65 72 61 74 65 |..\end{e|numerate|
|00000eb0| 7d 0d 0a 0d 0a 5c 73 65 | 63 74 69 6f 6e 7b 53 6f |}....\se|ction{So|
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|00000ef0| 5c 69 74 65 6d 20 20 5c | 6c 61 62 65 6c 7b 61 31 |\item  \|label{a1|
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|00000f20| 31 30 30 24 20 61 6e 64 | 20 24 31 32 30 24 20 62 |100$ and| $120$ b|
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|00000f70| 61 6c 6c 6f 77 62 72 65 | 61 6b 20 31 30 31 24 0d |allowbre|ak 101$.|
|00000f80| 0a 0d 0a 5c 69 74 65 6d | 20 20 5c 74 65 78 74 73 |...\item|  \texts|
|00000f90| 66 7b 46 61 63 74 6f 72 | 7d 20 24 31 30 33 3d 5c |f{Factor|} $103=\|
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|00000fb0| 0a 0d 0a 5c 69 74 65 6d | 20 20 5c 74 65 78 74 73 |...\item|  \texts|
|00000fc0| 66 7b 46 61 63 74 6f 72 | 7d 20 24 31 30 35 3d 5c |f{Factor|} $105=\|
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|00000ff0| 0a 5c 69 74 65 6d 20 20 | 5c 74 65 78 74 73 66 7b |.\item  |\textsf{|
|00001000| 46 61 63 74 6f 72 7d 20 | 24 31 30 37 3d 5c 61 6c |Factor} |$107=\al|
|00001010| 6c 6f 77 62 72 65 61 6b | 20 31 30 37 24 0d 0a 0d |lowbreak| 107$...|
|00001020| 0a 5c 69 74 65 6d 20 20 | 5c 74 65 78 74 73 66 7b |.\item  |\textsf{|
|00001030| 46 61 63 74 6f 72 7d 20 | 24 31 30 39 3d 5c 61 6c |Factor} |$109=\al|
|00001040| 6c 6f 77 62 72 65 61 6b | 20 31 30 39 24 0d 0a 0d |lowbreak| 109$...|
|00001050| 0a 5c 69 74 65 6d 20 20 | 5c 74 65 78 74 73 66 7b |.\item  |\textsf{|
|00001060| 46 61 63 74 6f 72 7d 5c | 74 65 78 74 73 63 7b 5c |Factor}\|textsc{\|
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|00001090| 0d 0a 5c 69 74 65 6d 20 | 20 5c 74 65 78 74 73 66 |..\item | \textsf|
|000010a0| 7b 46 61 63 74 6f 72 7d | 20 24 31 31 33 3d 5c 61 |{Factor}| $113=\a|
|000010b0| 6c 6c 6f 77 62 72 65 61 | 6b 20 31 31 33 24 0d 0a |llowbrea|k 113$..|
|000010c0| 0d 0a 5c 69 74 65 6d 20 | 20 5c 74 65 78 74 73 66 |..\item | \textsf|
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|000010e0| 6c 6c 6f 77 62 72 65 61 | 6b 20 35 5c 74 69 6d 65 |llowbrea|k 5\time|
|000010f0| 73 20 32 33 24 0d 0a 0d | 0a 5c 69 74 65 6d 20 20 |s 23$...|.\item  |
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|00001110| 24 31 31 37 3d 5c 61 6c | 6c 6f 77 62 72 65 61 6b |$117=\al|lowbreak|
|00001120| 20 33 5e 7b 32 7d 31 33 | 24 0d 0a 0d 0a 5c 69 74 | 3^{2}13|$....\it|
|00001130| 65 6d 20 20 5c 74 65 78 | 74 73 66 7b 46 61 63 74 |em  \tex|tsf{Fact|
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|00001160| 0d 0a 5c 65 6e 64 7b 65 | 6e 75 6d 65 72 61 74 65 |..\end{e|numerate|
|00001170| 7d 0d 0a 0d 0a 49 74 20 | 66 6f 6c 6c 6f 77 73 20 |}....It |follows |
|00001180| 74 68 61 74 20 74 68 65 | 20 70 72 69 6d 65 73 20 |that the| primes |
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|000011a0| 2c 20 24 31 30 37 24 2c | 20 24 31 30 39 24 2c 20 |, $107$,| $109$, |
|000011b0| 61 6e 64 20 24 31 31 33 | 24 2e 5c 68 66 69 6c 6c |and $113|$.\hfill|
|000011c0| 20 0d 0a 5c 68 79 70 65 | 72 72 65 66 7b 5c 54 43 | ..\hype|rref{\TC|
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|000011f0| 31 34 35 69 6e 7d 7b 30 | 2e 32 36 30 33 69 6e 7d |145in}{0|.2603in}|
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|00001210| 0d 0a 5c 69 74 65 6d 20 | 20 5c 6c 61 62 65 6c 7b |..\item | \label{|
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|00001250| 62 72 65 61 6b 20 34 33 | 5c 66 72 61 63 7b 25 0d |break 43|\frac{%.|
|00001260| 0a 31 31 7d 7b 32 33 7d | 24 2e 20 4e 6f 74 65 20 |.11}{23}|$. Note |
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|000012e0| 61 6c 6c 6f 77 62 72 65 | 61 6b 20 32 33 24 25 0d |allowbre|ak 23$%.|
|000012f0| 0a 2e 5c 68 66 69 6c 6c | 20 5c 68 79 70 65 72 72 |..\hfill| \hyperr|
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|00001350| 20 20 5c 6c 61 62 65 6c | 7b 61 33 7d 4e 6f 74 65 |  \label|{a3}Note|
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|00001380| 66 7b 45 76 61 6c 75 61 | 74 65 7d 20 79 69 65 6c |f{Evalua|te} yiel|
|00001390| 64 20 0d 0a 5c 62 65 67 | 69 6e 7b 65 71 6e 61 72 |d ..\beg|in{eqnar|
|000013a0| 72 61 79 2a 7d 0d 0a 31 | 39 21 20 26 3d 26 32 5e |ray*}..1|9! &=&2^|
|000013b0| 7b 31 36 7d 33 5e 7b 38 | 7d 35 5e 7b 33 7d 37 5e |{16}3^{8|}5^{3}7^|
|000013c0| 7b 32 7d 31 31 5c 74 69 | 6d 65 73 20 31 33 5c 74 |{2}11\ti|mes 13\t|
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|000013e0| 5c 61 6c 6c 6f 77 62 72 | 65 61 6b 0d 0a 3d 31 32 |\allowbr|eak..=12|
|000013f0| 31 5c 2c 36 34 35 5c 2c | 31 30 30 5c 2c 5c 61 6c |1\,645\,|100\,\al|
|00001400| 6c 6f 77 62 72 65 61 6b | 20 34 30 38 5c 2c 38 33 |lowbreak| 408\,83|
|00001410| 32 5c 2c 30 30 30 20 5c | 5c 0d 0a 32 30 21 20 26 |2\,000 \|\..20! &|
|00001420| 3d 26 32 5e 7b 31 38 7d | 33 5e 7b 38 7d 35 5e 7b |=&2^{18}|3^{8}5^{|
|00001430| 34 7d 37 5e 7b 32 7d 31 | 31 5c 74 69 6d 65 73 20 |4}7^{2}1|1\times |
|00001440| 31 33 5c 74 69 6d 65 73 | 20 31 37 5c 74 69 6d 65 |13\times| 17\time|
|00001450| 73 20 31 39 5c 61 6c 6c | 6f 77 62 72 65 61 6b 0d |s 19\all|owbreak.|
|00001460| 0a 3d 32 34 33 32 5c 2c | 39 30 32 5c 2c 30 30 38 |.=2432\,|902\,008|
|00001470| 5c 2c 5c 61 6c 6c 6f 77 | 62 72 65 61 6b 20 31 37 |\,\allow|break 17|
|00001480| 36 5c 2c 36 34 30 5c 2c | 30 30 30 20 5c 5c 0d 0a |6\,640\,|000 \\..|
|00001490| 32 31 21 20 26 3d 26 5c | 61 6c 6c 6f 77 62 72 65 |21! &=&\|allowbre|
|000014a0| 61 6b 20 32 5e 7b 31 38 | 7d 33 5e 7b 39 7d 35 5e |ak 2^{18|}3^{9}5^|
|000014b0| 7b 34 7d 37 5e 7b 33 7d | 31 31 5c 74 69 6d 65 73 |{4}7^{3}|11\times|
|000014c0| 20 31 33 5c 74 69 6d 65 | 73 20 31 37 5c 74 69 6d | 13\time|s 17\tim|
|000014d0| 65 73 0d 0a 31 39 3d 5c | 61 6c 6c 6f 77 62 72 65 |es..19=\|allowbre|
|000014e0| 61 6b 20 35 31 5c 2c 5c | 61 6c 6c 6f 77 62 72 65 |ak 51\,\|allowbre|
|000014f0| 61 6b 20 30 39 30 5c 2c | 39 34 32 5c 2c 31 37 31 |ak 090\,|942\,171|
|00001500| 5c 2c 5c 61 6c 6c 6f 77 | 62 72 65 61 6b 20 37 30 |\,\allow|break 70|
|00001510| 39 5c 2c 34 34 30 5c 2c | 30 30 30 20 5c 5c 0d 0a |9\,440\,|000 \\..|
|00001520| 32 32 21 20 26 3d 26 32 | 5e 7b 31 39 7d 33 5e 7b |22! &=&2|^{19}3^{|
|00001530| 39 7d 35 5e 7b 34 7d 37 | 5e 7b 33 7d 31 31 5e 7b |9}5^{4}7|^{3}11^{|
|00001540| 32 7d 31 33 5c 74 69 6d | 65 73 20 31 37 5c 74 69 |2}13\tim|es 17\ti|
|00001550| 6d 65 73 20 31 39 3d 5c | 61 6c 6c 6f 77 62 72 65 |mes 19=\|allowbre|
|00001560| 61 6b 0d 0a 31 31 32 34 | 5c 2c 5c 61 6c 6c 6f 77 |ak..1124|\,\allow|
|00001570| 62 72 65 61 6b 20 30 30 | 30 5c 2c 37 32 37 5c 2c |break 00|0\,727\,|
|00001580| 37 37 37 5c 2c 5c 61 6c | 6c 6f 77 62 72 65 61 6b |777\,\al|lowbreak|
|00001590| 20 36 30 37 5c 2c 36 38 | 30 5c 2c 30 30 30 20 5c | 607\,68|0\,000 \|
|000015a0| 5c 0d 0a 32 33 21 20 26 | 3d 26 32 5e 7b 31 39 7d |\..23! &|=&2^{19}|
|000015b0| 33 5e 7b 39 7d 35 5e 7b | 34 7d 37 5e 7b 33 7d 31 |3^{9}5^{|4}7^{3}1|
|000015c0| 31 5e 7b 32 7d 31 33 5c | 74 69 6d 65 73 20 31 37 |1^{2}13\|times 17|
|000015d0| 5c 74 69 6d 65 73 20 31 | 39 5c 74 69 6d 65 73 20 |\times 1|9\times |
|000015e0| 32 33 3d 5c 61 6c 6c 6f | 77 62 72 65 61 6b 0d 0a |23=\allo|wbreak..|
|000015f0| 32 35 5c 2c 38 35 32 5c | 2c 5c 61 6c 6c 6f 77 62 |25\,852\|,\allowb|
|00001600| 72 65 61 6b 20 30 31 36 | 5c 2c 37 33 38 5c 2c 38 |reak 016|\,738\,8|
|00001610| 38 34 5c 2c 5c 61 6c 6c | 6f 77 62 72 65 61 6b 20 |84\,\all|owbreak |
|00001620| 39 37 36 5c 2c 36 34 30 | 5c 2c 30 30 30 20 5c 5c |976\,640|\,000 \\|
|00001630| 0d 0a 32 34 21 20 26 3d | 26 32 5e 7b 32 32 7d 33 |..24! &=|&2^{22}3|
|00001640| 5e 7b 31 30 7d 35 5e 7b | 34 7d 37 5e 7b 33 7d 31 |^{10}5^{|4}7^{3}1|
|00001650| 31 5e 7b 32 7d 31 33 5c | 74 69 6d 65 73 20 31 37 |1^{2}13\|times 17|
|00001660| 5c 74 69 6d 65 73 20 31 | 39 5c 74 69 6d 65 73 20 |\times 1|9\times |
|00001670| 32 33 3d 5c 61 6c 6c 6f | 77 62 72 65 61 6b 0d 0a |23=\allo|wbreak..|
|00001680| 36 32 30 5c 2c 34 34 38 | 5c 2c 5c 61 6c 6c 6f 77 |620\,448|\,\allow|
|00001690| 62 72 65 61 6b 20 34 30 | 31 5c 2c 37 33 33 5c 2c |break 40|1\,733\,|
|000016a0| 32 33 39 5c 2c 5c 61 6c | 6c 6f 77 62 72 65 61 6b |239\,\al|lowbreak|
|000016b0| 20 34 33 39 5c 2c 33 36 | 30 5c 2c 30 30 30 20 5c | 439\,36|0\,000 \|
|000016c0| 5c 0d 0a 32 35 21 20 26 | 3d 26 5c 61 6c 6c 6f 77 |\..25! &|=&\allow|
|000016d0| 62 72 65 61 6b 20 32 5e | 7b 32 32 7d 33 5e 7b 31 |break 2^|{22}3^{1|
|000016e0| 30 7d 35 5e 7b 36 7d 37 | 5e 7b 33 7d 31 31 5e 7b |0}5^{6}7|^{3}11^{|
|000016f0| 32 7d 31 33 5c 74 69 6d | 65 73 20 31 37 5c 74 69 |2}13\tim|es 17\ti|
|00001700| 6d 65 73 20 31 39 5c 74 | 69 6d 65 73 0d 0a 32 33 |mes 19\t|imes..23|
|00001710| 3d 5c 61 6c 6c 6f 77 62 | 72 65 61 6b 20 31 35 5c |=\allowb|reak 15\|
|00001720| 2c 35 31 31 5c 2c 32 31 | 30 5c 2c 5c 61 6c 6c 6f |,511\,21|0\,\allo|
|00001730| 77 62 72 65 61 6b 20 30 | 34 33 5c 2c 33 33 30 5c |wbreak 0|43\,330\|
|00001740| 2c 39 38 35 5c 2c 5c 61 | 6c 6c 6f 77 62 72 65 61 |,985\,\a|llowbrea|
|00001750| 6b 0d 0a 39 38 34 5c 2c | 30 30 30 5c 2c 30 30 30 |k..984\,|000\,000|
|00001760| 0d 0a 5c 65 6e 64 7b 65 | 71 6e 61 72 72 61 79 2a |..\end{e|qnarray*|
|00001770| 7d 0d 0a 61 6e 64 20 68 | 65 6e 63 65 20 24 31 39 |}..and h|ence $19|
|00001780| 21 24 20 65 6e 64 73 20 | 69 6e 20 24 33 24 20 7a |!$ ends |in $3$ z|
|00001790| 65 72 6f 73 2c 20 24 32 | 30 21 24 20 65 6e 64 73 |eros, $2|0!$ ends|
|000017a0| 20 69 6e 20 24 34 24 2c | 20 61 6e 64 20 24 32 35 | in $4$,| and $25|
|000017b0| 21 24 20 65 6e 64 73 20 | 69 6e 20 24 36 24 0d 0a |!$ ends |in $6$..|
|000017c0| 7a 65 72 6f 73 2e 20 49 | 74 20 61 70 70 65 61 72 |zeros. I|t appear|
|000017d0| 73 20 74 68 61 74 20 74 | 68 65 20 65 78 70 6f 6e |s that t|he expon|
|000017e0| 65 6e 74 20 6f 66 20 24 | 35 24 20 63 6f 75 6e 74 |ent of $|5$ count|
|000017f0| 73 20 74 68 65 20 6e 75 | 6d 62 65 72 20 6f 66 20 |s the nu|mber of |
|00001800| 74 72 61 69 6c 69 6e 67 | 0d 0a 7a 65 72 6f 73 2e |trailing|..zeros.|
|00001810| 20 54 68 65 20 6e 75 6d | 62 65 72 20 6f 66 20 66 | The num|ber of f|
|00001820| 69 76 65 73 20 69 6e 20 | 24 31 30 30 21 24 20 69 |ives in |$100!$ i|
|00001830| 73 20 67 69 76 65 6e 20 | 62 79 20 24 5c 66 72 61 |s given |by $\fra|
|00001840| 63 7b 31 30 30 7d 7b 35 | 7d 2b 5c 66 72 61 63 7b |c{100}{5|}+\frac{|
|00001850| 31 30 30 7d 7b 32 35 25 | 0d 0a 7d 3d 5c 61 6c 6c |100}{25%|..}=\all|
|00001860| 6f 77 62 72 65 61 6b 20 | 32 34 24 2c 20 61 6e 64 |owbreak |24$, and|
|00001870| 20 64 69 72 65 63 74 20 | 65 76 61 6c 75 61 74 69 | direct |evaluati|
|00001880| 6f 6e 20 73 68 6f 77 73 | 20 74 68 61 74 20 24 31 |on shows| that $1|
|00001890| 30 30 21 24 20 65 6e 64 | 73 20 69 6e 20 24 32 34 |00!$ end|s in $24|
|000018a0| 24 0d 0a 7a 65 72 6f 73 | 2e 20 49 6e 20 67 65 6e |$..zeros|. In gen|
|000018b0| 65 72 61 6c 2c 20 74 68 | 65 20 6e 75 6d 62 65 72 |eral, th|e number|
|000018c0| 20 6f 66 20 74 72 61 69 | 6c 69 6e 67 20 7a 65 72 | of trai|ling zer|
|000018d0| 6f 73 20 69 6e 20 24 6e | 21 24 20 69 73 20 67 69 |os in $n|!$ is gi|
|000018e0| 76 65 6e 20 62 79 20 74 | 68 65 20 73 75 6d 20 0d |ven by t|he sum .|
|000018f0| 0a 5c 5b 0d 0a 5c 6c 65 | 66 74 5c 6c 66 6c 6f 6f |.\[..\le|ft\lfloo|
|00001900| 72 20 5c 66 72 61 63 7b | 6e 7d 7b 35 7d 5c 72 69 |r \frac{|n}{5}\ri|
|00001910| 67 68 74 5c 72 66 6c 6f | 6f 72 20 2b 5c 6c 65 66 |ght\rflo|or +\lef|
|00001920| 74 5c 6c 66 6c 6f 6f 72 | 20 5c 66 72 61 63 7b 6e |t\lfloor| \frac{n|
|00001930| 7d 7b 35 5e 7b 32 7d 7d | 25 0d 0a 5c 72 69 67 68 |}{5^{2}}|%..\righ|
|00001940| 74 5c 72 66 6c 6f 6f 72 | 20 2b 5c 6c 65 66 74 5c |t\rfloor| +\left\|
|00001950| 6c 66 6c 6f 6f 72 20 5c | 66 72 61 63 7b 6e 7d 7b |lfloor \|frac{n}{|
|00001960| 35 5e 7b 33 7d 7d 5c 72 | 69 67 68 74 5c 72 66 6c |5^{3}}\r|ight\rfl|
|00001970| 6f 6f 72 20 2b 5c 6c 65 | 66 74 5c 6c 66 6c 6f 6f |oor +\le|ft\lfloo|
|00001980| 72 20 5c 66 72 61 63 7b | 25 0d 0a 6e 7d 7b 35 5e |r \frac{|%..n}{5^|
|00001990| 7b 34 7d 7d 5c 72 69 67 | 68 74 5c 72 66 6c 6f 6f |{4}}\rig|ht\rfloo|
|000019a0| 72 20 2b 5c 63 64 6f 74 | 73 20 0d 0a 5c 5d 0d 0a |r +\cdot|s ..\]..|
|000019b0| 77 68 65 72 65 20 24 5c | 6c 65 66 74 5c 6c 66 6c |where $\|left\lfl|
|000019c0| 6f 6f 72 20 78 5c 72 69 | 67 68 74 5c 72 66 6c 6f |oor x\ri|ght\rflo|
|000019d0| 6f 72 20 24 20 64 65 6e | 6f 74 65 73 20 74 68 65 |or $ den|otes the|
|000019e0| 20 67 72 65 61 74 65 73 | 74 20 69 6e 74 65 67 65 | greates|t intege|
|000019f0| 72 20 24 5c 6c 65 71 20 | 78 24 2e 0d 0a 49 6e 20 |r $\leq |x$...In |
|00001a00| 70 61 72 74 69 63 75 6c | 61 72 2c 20 0d 0a 5c 62 |particul|ar, ..\b|
|00001a10| 65 67 69 6e 7b 65 71 6e | 61 72 72 61 79 2a 7d 0d |egin{eqn|array*}.|
|00001a20| 0a 5c 6c 65 66 74 5c 6c | 66 6c 6f 6f 72 20 5c 66 |.\left\l|floor \f|
|00001a30| 72 61 63 7b 31 32 35 7d | 7b 35 7d 5c 72 69 67 68 |rac{125}|{5}\righ|
|00001a40| 74 5c 72 66 6c 6f 6f 72 | 20 2b 5c 6c 65 66 74 5c |t\rfloor| +\left\|
|00001a50| 6c 66 6c 6f 6f 72 20 5c | 66 72 61 63 7b 31 32 35 |lfloor \|frac{125|
|00001a60| 7d 7b 35 5e 7b 32 7d 7d | 25 0d 0a 5c 72 69 67 68 |}{5^{2}}|%..\righ|
|00001a70| 74 5c 72 66 6c 6f 6f 72 | 20 2b 5c 6c 65 66 74 5c |t\rfloor| +\left\|
|00001a80| 6c 66 6c 6f 6f 72 20 5c | 66 72 61 63 7b 31 32 35 |lfloor \|frac{125|
|00001a90| 7d 7b 35 5e 7b 33 7d 7d | 5c 72 69 67 68 74 5c 72 |}{5^{3}}|\right\r|
|00001aa0| 66 6c 6f 6f 72 20 26 3d | 26 32 35 2b 35 2b 31 3d |floor &=|&25+5+1=|
|00001ab0| 33 31 20 5c 5c 0d 0a 5c | 6c 65 66 74 5c 6c 66 6c |31 \\..\|left\lfl|
|00001ac0| 6f 6f 72 20 5c 66 72 61 | 63 7b 32 30 30 7d 7b 35 |oor \fra|c{200}{5|
|00001ad0| 7d 5c 72 69 67 68 74 5c | 72 66 6c 6f 6f 72 20 2b |}\right\|rfloor +|
|00001ae0| 5c 6c 65 66 74 5c 6c 66 | 6c 6f 6f 72 20 5c 66 72 |\left\lf|loor \fr|
|00001af0| 61 63 7b 32 30 30 7d 7b | 35 5e 7b 32 7d 7d 25 0d |ac{200}{|5^{2}}%.|
|00001b00| 0a 5c 72 69 67 68 74 5c | 72 66 6c 6f 6f 72 20 2b |.\right\|rfloor +|
|00001b10| 5c 6c 65 66 74 5c 6c 66 | 6c 6f 6f 72 20 5c 66 72 |\left\lf|loor \fr|
|00001b20| 61 63 7b 32 30 30 7d 7b | 35 5e 7b 33 7d 7d 5c 72 |ac{200}{|5^{3}}\r|
|00001b30| 69 67 68 74 5c 72 66 6c | 6f 6f 72 20 26 3d 26 34 |ight\rfl|oor &=&4|
|00001b40| 30 2b 38 2b 31 3d 34 39 | 20 5c 5c 0d 0a 5c 6c 65 |0+8+1=49| \\..\le|
|00001b50| 66 74 5c 6c 66 6c 6f 6f | 72 20 5c 66 72 61 63 7b |ft\lfloo|r \frac{|
|00001b60| 35 30 30 7d 7b 35 7d 5c | 72 69 67 68 74 5c 72 66 |500}{5}\|right\rf|
|00001b70| 6c 6f 6f 72 20 2b 5c 6c | 65 66 74 5c 6c 66 6c 6f |loor +\l|eft\lflo|
|00001b80| 6f 72 20 5c 66 72 61 63 | 7b 35 30 30 7d 7b 35 5e |or \frac|{500}{5^|
|00001b90| 7b 32 7d 7d 25 0d 0a 5c | 72 69 67 68 74 5c 72 66 |{2}}%..\|right\rf|
|00001ba0| 6c 6f 6f 72 20 2b 5c 6c | 65 66 74 5c 6c 66 6c 6f |loor +\l|eft\lflo|
|00001bb0| 6f 72 20 5c 66 72 61 63 | 7b 35 30 30 7d 7b 35 5e |or \frac|{500}{5^|
|00001bc0| 7b 33 7d 7d 5c 72 69 67 | 68 74 5c 72 66 6c 6f 6f |{3}}\rig|ht\rfloo|
|00001bd0| 72 20 26 3d 26 31 30 30 | 2b 32 30 2b 34 3d 31 32 |r &=&100|+20+4=12|
|00001be0| 34 20 5c 5c 0d 0a 5c 6c | 65 66 74 5c 6c 66 6c 6f |4 \\..\l|eft\lflo|
|00001bf0| 6f 72 20 5c 66 72 61 63 | 7b 36 32 35 7d 7b 35 7d |or \frac|{625}{5}|
|00001c00| 5c 72 69 67 68 74 5c 72 | 66 6c 6f 6f 72 20 2b 5c |\right\r|floor +\|
|00001c10| 6c 65 66 74 5c 6c 66 6c | 6f 6f 72 20 5c 66 72 61 |left\lfl|oor \fra|
|00001c20| 63 7b 36 32 35 7d 7b 35 | 5e 7b 32 7d 7d 25 0d 0a |c{625}{5|^{2}}%..|
|00001c30| 5c 72 69 67 68 74 5c 72 | 66 6c 6f 6f 72 20 2b 5c |\right\r|floor +\|
|00001c40| 6c 65 66 74 5c 6c 66 6c | 6f 6f 72 20 5c 66 72 61 |left\lfl|oor \fra|
|00001c50| 63 7b 36 32 35 7d 7b 35 | 5e 7b 33 7d 7d 5c 72 69 |c{625}{5|^{3}}\ri|
|00001c60| 67 68 74 5c 72 66 6c 6f | 6f 72 20 2b 5c 6c 65 66 |ght\rflo|or +\lef|
|00001c70| 74 5c 6c 66 6c 6f 6f 72 | 20 0d 0a 5c 66 72 61 63 |t\lfloor| ..\frac|
|00001c80| 7b 36 32 35 7d 7b 35 5e | 7b 34 7d 7d 5c 72 69 67 |{625}{5^|{4}}\rig|
|00001c90| 68 74 5c 72 66 6c 6f 6f | 72 20 26 3d 26 31 32 35 |ht\rfloo|r &=&125|
|00001ca0| 2b 32 35 2b 35 2b 31 3d | 31 35 36 20 5c 5c 0d 0a |+25+5+1=|156 \\..|
|00001cb0| 5c 6c 65 66 74 5c 6c 66 | 6c 6f 6f 72 20 5c 66 72 |\left\lf|loor \fr|
|00001cc0| 61 63 7b 31 30 30 30 7d | 7b 35 7d 5c 72 69 67 68 |ac{1000}|{5}\righ|
|00001cd0| 74 5c 72 66 6c 6f 6f 72 | 20 2b 5c 6c 65 66 74 5c |t\rfloor| +\left\|
|00001ce0| 6c 66 6c 6f 6f 72 20 5c | 66 72 61 63 7b 31 30 30 |lfloor \|frac{100|
|00001cf0| 30 7d 7b 35 5e 7b 32 7d | 7d 25 0d 0a 5c 72 69 67 |0}{5^{2}|}%..\rig|
|00001d00| 68 74 5c 72 66 6c 6f 6f | 72 20 2b 5c 6c 65 66 74 |ht\rfloo|r +\left|
|00001d10| 5c 6c 66 6c 6f 6f 72 20 | 5c 66 72 61 63 7b 31 30 |\lfloor |\frac{10|
|00001d20| 30 30 7d 7b 35 5e 7b 33 | 7d 7d 5c 72 69 67 68 74 |00}{5^{3|}}\right|
|00001d30| 5c 72 66 6c 6f 6f 72 20 | 2b 5c 6c 65 66 74 5c 6c |\rfloor |+\left\l|
|00001d40| 66 6c 6f 6f 72 20 0d 0a | 5c 66 72 61 63 7b 31 30 |floor ..|\frac{10|
|00001d50| 30 30 7d 7b 35 5e 7b 34 | 7d 7d 5c 72 69 67 68 74 |00}{5^{4|}}\right|
|00001d60| 5c 72 66 6c 6f 6f 72 20 | 26 3d 26 32 30 30 2b 34 |\rfloor |&=&200+4|
|00001d70| 30 2b 38 2b 31 3d 32 34 | 39 0d 0a 5c 65 6e 64 7b |0+8+1=24|9..\end{|
|00001d80| 65 71 6e 61 72 72 61 79 | 2a 7d 0d 0a 47 6f 20 61 |eqnarray|*}..Go a|
|00001d90| 68 65 61 64 20 61 6e 64 | 20 76 65 72 69 66 79 20 |head and| verify |
|00001da0| 74 68 61 74 20 74 68 65 | 73 65 20 63 6f 75 6e 74 |that the|se count|
|00001db0| 20 74 68 65 20 6e 75 6d | 62 65 72 20 6f 66 20 74 | the num|ber of t|
|00001dc0| 72 61 69 6c 69 6e 67 20 | 7a 65 72 6f 73 20 69 6e |railing |zeros in|
|00001dd0| 20 24 31 32 35 21 24 2c | 20 0d 0a 24 32 30 30 21 | $125!$,| ..$200!|
|00001de0| 24 2c 20 24 35 30 30 21 | 24 2c 20 24 36 32 35 21 |$, $500!|$, $625!|
|00001df0| 24 2c 20 61 6e 64 20 24 | 31 30 30 30 21 2c 24 20 |$, and $|1000!,$ |
|00001e00| 72 65 73 70 65 63 74 69 | 76 65 6c 79 2e 20 49 74 |respecti|vely. It|
|00001e10| 20 74 61 6b 65 73 20 72 | 6f 75 67 68 6c 79 0d 0a | takes r|oughly..|
|00001e20| 74 68 72 65 65 2d 71 75 | 61 72 74 65 72 73 20 6f |three-qu|arters o|
|00001e30| 66 20 61 20 31 37 2d 69 | 6e 63 68 20 63 6f 6d 70 |f a 17-i|nch comp|
|00001e40| 75 74 65 72 20 73 63 72 | 65 65 6e 20 74 6f 20 64 |uter scr|een to d|
|00001e50| 69 73 70 6c 61 79 20 24 | 31 30 30 30 21 24 2c 20 |isplay $|1000!$, |
|00001e60| 61 6e 64 20 74 68 65 0d | 0a 72 65 73 75 6c 74 73 |and the.|.results|
|00001e70| 20 61 72 65 20 6e 6f 74 | 20 64 69 73 70 6c 61 79 | are not| display|
|00001e80| 65 64 20 68 65 72 65 2e | 5c 68 66 69 6c 6c 20 5c |ed here.|\hfill \|
|00001e90| 68 79 70 65 72 72 65 66 | 7b 5c 54 43 49 49 63 6f |hyperref|{\TCIIco|
|00001ea0| 6e 7b 25 0d 0a 42 49 54 | 4d 41 50 53 45 54 50 72 |n{%..BIT|MAPSETPr|
|00001eb0| 6f 62 53 6f 6c 76 48 69 | 6e 74 7d 7b 30 2e 32 31 |obSolvHi|nt}{0.21|
|00001ec0| 34 35 69 6e 7d 7b 30 2e | 32 36 30 33 69 6e 7d 7b |45in}{0.|2603in}{|
|00001ed0| 30 69 6e 7d 7d 7b 7d 7b | 7d 7b 71 33 7d 0d 0a 0d |0in}}{}{|}{q3}...|
|00001ee0| 0a 5c 69 74 65 6d 20 20 | 5c 6c 61 62 65 6c 7b 61 |.\item  |\label{a|
|00001ef0| 34 7d 4e 6f 74 65 20 74 | 68 61 74 20 0d 0a 5c 62 |4}Note t|hat ..\b|
|00001f00| 65 67 69 6e 7b 65 71 6e | 61 72 72 61 79 2a 7d 0d |egin{eqn|array*}.|
|00001f10| 0a 5c 6c 65 66 74 28 20 | 31 2b 5c 66 72 61 63 7b |.\left( |1+\frac{|
|00001f20| 31 7d 7b 32 7d 5c 72 69 | 67 68 74 29 20 5e 7b 32 |1}{2}\ri|ght) ^{2|
|00001f30| 7d 20 26 3d 26 5c 61 6c | 6c 6f 77 62 72 65 61 6b |} &=&\al|lowbreak|
|00001f40| 20 32 2e 5c 2c 5c 61 6c | 6c 6f 77 62 72 65 61 6b | 2.\,\al|lowbreak|
|00001f50| 20 32 35 20 5c 5c 0d 0a | 5c 6c 65 66 74 28 20 31 | 25 \\..|\left( 1|
|00001f60| 2b 5c 66 72 61 63 7b 31 | 7d 7b 34 7d 5c 72 69 67 |+\frac{1|}{4}\rig|
|00001f70| 68 74 29 20 5e 7b 34 7d | 20 26 3d 26 5c 61 6c 6c |ht) ^{4}| &=&\all|
|00001f80| 6f 77 62 72 65 61 6b 20 | 32 2e 5c 2c 5c 61 6c 6c |owbreak |2.\,\all|
|00001f90| 6f 77 62 72 65 61 6b 20 | 34 34 31 5c 2c 34 30 36 |owbreak |441\,406|
|00001fa0| 5c 2c 32 35 0d 0a 5c 5c | 0d 0a 5c 6c 65 66 74 28 |\,25..\\|..\left(|
|00001fb0| 20 31 2b 5c 66 72 61 63 | 7b 31 7d 7b 38 7d 5c 72 | 1+\frac|{1}{8}\r|
|00001fc0| 69 67 68 74 29 20 5e 7b | 38 7d 20 26 3d 26 5c 61 |ight) ^{|8} &=&\a|
|00001fd0| 6c 6c 6f 77 62 72 65 61 | 6b 20 32 2e 5c 2c 5c 61 |llowbrea|k 2.\,\a|
|00001fe0| 6c 6c 6f 77 62 72 65 61 | 6b 0d 0a 35 36 35 5c 2c |llowbrea|k..565\,|
|00001ff0| 37 38 34 5c 2c 35 31 33 | 5c 2c 5c 61 6c 6c 6f 77 |784\,513|\,\allow|
|00002000| 62 72 65 61 6b 20 39 35 | 30 5c 2c 33 34 37 5c 2c |break 95|0\,347\,|
|00002010| 39 30 30 5c 2c 5c 61 6c | 6c 6f 77 62 72 65 61 6b |900\,\al|lowbreak|
|00002020| 20 34 20 5c 5c 0d 0a 5c | 6c 65 66 74 28 20 31 2b | 4 \\..\|left( 1+|
|00002030| 5c 66 72 61 63 7b 31 7d | 7b 31 36 7d 5c 72 69 67 |\frac{1}|{16}\rig|
|00002040| 68 74 29 20 5e 7b 31 36 | 7d 20 26 3d 26 5c 61 6c |ht) ^{16|} &=&\al|
|00002050| 6c 6f 77 62 72 65 61 6b | 20 32 2e 5c 2c 5c 61 6c |lowbreak| 2.\,\al|
|00002060| 6c 6f 77 62 72 65 61 6b | 0d 0a 36 33 37 5c 2c 39 |lowbreak|..637\,9|
|00002070| 32 38 5c 2c 34 39 37 5c | 2c 5c 61 6c 6c 6f 77 62 |28\,497\|,\allowb|
|00002080| 72 65 61 6b 20 33 36 36 | 5c 2c 35 39 39 5c 2c 38 |reak 366|\,599\,8|
|00002090| 35 38 5c 2c 5c 61 6c 6c | 6f 77 62 72 65 61 6b 20 |58\,\all|owbreak |
|000020a0| 38 20 5c 5c 0d 0a 5c 6c | 65 66 74 28 20 31 2b 5c |8 \\..\l|eft( 1+\|
|000020b0| 66 72 61 63 7b 31 7d 7b | 33 32 7d 5c 72 69 67 68 |frac{1}{|32}\righ|
|000020c0| 74 29 20 5e 7b 33 32 7d | 20 26 3d 26 5c 61 6c 6c |t) ^{32}| &=&\all|
|000020d0| 6f 77 62 72 65 61 6b 20 | 32 2e 5c 2c 5c 61 6c 6c |owbreak |2.\,\all|
|000020e0| 6f 77 62 72 65 61 6b 0d | 0a 36 37 36 5c 2c 39 39 |owbreak.|.676\,99|
|000020f0| 30 5c 2c 31 32 39 5c 2c | 5c 61 6c 6c 6f 77 62 72 |0\,129\,|\allowbr|
|00002100| 65 61 6b 20 33 37 38 5c | 2c 31 38 32 5c 2c 36 38 |eak 378\|,182\,68|
|00002110| 34 5c 2c 5c 61 6c 6c 6f | 77 62 72 65 61 6b 20 36 |4\,\allo|wbreak 6|
|00002120| 20 5c 5c 0d 0a 5c 6c 65 | 66 74 28 20 31 2b 5c 66 | \\..\le|ft( 1+\f|
|00002130| 72 61 63 7b 31 7d 7b 36 | 34 7d 5c 72 69 67 68 74 |rac{1}{6|4}\right|
|00002140| 29 20 5e 7b 36 34 7d 20 | 26 3d 26 5c 61 6c 6c 6f |) ^{64} |&=&\allo|
|00002150| 77 62 72 65 61 6b 20 32 | 2e 5c 2c 5c 61 6c 6c 6f |wbreak 2|.\,\allo|
|00002160| 77 62 72 65 61 6b 0d 0a | 36 39 37 5c 2c 33 34 34 |wbreak..|697\,344|
|00002170| 5c 2c 39 35 32 5c 2c 5c | 61 6c 6c 6f 77 62 72 65 |\,952\,\|allowbre|
|00002180| 61 6b 20 35 36 35 5c 2c | 30 39 38 5c 2c 38 35 35 |ak 565\,|098\,855|
|00002190| 5c 2c 5c 61 6c 6c 6f 77 | 62 72 65 61 6b 20 36 20 |\,\allow|break 6 |
|000021a0| 5c 5c 0d 0a 5c 6c 65 66 | 74 28 20 31 2b 5c 66 72 |\\..\lef|t( 1+\fr|
|000021b0| 61 63 7b 31 7d 7b 31 32 | 38 7d 5c 72 69 67 68 74 |ac{1}{12|8}\right|
|000021c0| 29 20 5e 7b 31 32 38 7d | 20 26 3d 26 5c 61 6c 6c |) ^{128}| &=&\all|
|000021d0| 6f 77 62 72 65 61 6b 20 | 32 2e 5c 2c 5c 61 6c 6c |owbreak |2.\,\all|
|000021e0| 6f 77 62 72 65 61 6b 0d | 0a 37 30 37 5c 2c 37 33 |owbreak.|.707\,73|
|000021f0| 39 5c 2c 30 31 39 5c 2c | 5c 61 6c 6c 6f 77 62 72 |9\,019\,|\allowbr|
|00002200| 65 61 6b 20 36 38 38 5c | 2c 30 32 30 5c 2c 34 39 |eak 688\|,020\,49|
|00002210| 32 20 5c 5c 0d 0a 5c 6c | 65 66 74 28 20 31 2b 5c |2 \\..\l|eft( 1+\|
|00002220| 66 72 61 63 7b 31 7d 7b | 32 35 36 7d 5c 72 69 67 |frac{1}{|256}\rig|
|00002230| 68 74 29 20 5e 7b 32 35 | 36 7d 20 26 3d 26 5c 61 |ht) ^{25|6} &=&\a|
|00002240| 6c 6c 6f 77 62 72 65 61 | 6b 20 32 2e 5c 2c 5c 61 |llowbrea|k 2.\,\a|
|00002250| 6c 6c 6f 77 62 72 65 61 | 6b 0d 0a 37 31 32 5c 2c |llowbrea|k..712\,|
|00002260| 39 39 31 5c 2c 36 32 34 | 5c 2c 5c 61 6c 6c 6f 77 |991\,624|\,\allow|
|00002270| 62 72 65 61 6b 20 32 35 | 33 5c 2c 34 33 34 5c 2c |break 25|3\,434\,|
|00002280| 33 33 34 5c 2c 5c 61 6c | 6c 6f 77 62 72 65 61 6b |334\,\al|lowbreak|
|00002290| 20 33 0d 0a 5c 65 6e 64 | 7b 65 71 6e 61 72 72 61 | 3..\end|{eqnarra|
|000022a0| 79 2a 7d 0d 0a 54 68 65 | 20 6e 75 6d 62 65 72 20 |y*}..The| number |
|000022b0| 24 65 3d 5c 61 6c 6c 6f | 77 62 72 65 61 6b 20 32 |$e=\allo|wbreak 2|
|000022c0| 2e 5c 2c 5c 61 6c 6c 6f | 77 62 72 65 61 6b 20 37 |.\,\allo|wbreak 7|
|000022d0| 31 38 5c 2c 32 38 31 5c | 2c 38 32 38 5c 2c 5c 61 |18\,281\|,828\,\a|
|000022e0| 6c 6c 6f 77 62 72 65 61 | 6b 0d 0a 34 35 39 5c 2c |llowbrea|k..459\,|
|000022f0| 30 34 35 5c 2c 32 33 35 | 5c 2c 5c 61 6c 6c 6f 77 |045\,235|\,\allow|
|00002300| 62 72 65 61 6b 20 34 24 | 20 69 73 20 62 65 67 69 |break 4$| is begi|
|00002310| 6e 6e 69 6e 67 20 74 6f | 20 65 6d 65 72 67 65 2e |nning to| emerge.|
|00002320| 5c 68 66 69 6c 6c 20 5c | 68 79 70 65 72 72 65 66 |\hfill \|hyperref|
|00002330| 7b 25 0d 0a 5c 54 43 49 | 49 63 6f 6e 7b 42 49 54 |{%..\TCI|Icon{BIT|
|00002340| 4d 41 50 53 45 54 50 72 | 6f 62 53 6f 6c 76 48 69 |MAPSETPr|obSolvHi|
|00002350| 6e 74 7d 7b 30 2e 32 31 | 34 35 69 6e 7d 7b 30 2e |nt}{0.21|45in}{0.|
|00002360| 32 36 30 33 69 6e 7d 7b | 30 69 6e 7d 7d 7b 7d 7b |2603in}{|0in}}{}{|
|00002370| 7d 7b 71 34 7d 0d 0a 0d | 0a 5c 69 74 65 6d 20 20 |}{q4}...|.\item  |
|00002380| 5c 6c 61 62 65 6c 7b 61 | 35 7d 57 69 74 68 20 74 |\label{a|5}With t|
|00002390| 68 65 20 6e 75 6d 62 65 | 72 73 20 24 31 24 2c 20 |he numbe|rs $1$, |
|000023a0| 24 32 24 2c 20 61 6e 64 | 20 24 33 24 20 77 65 20 |$2$, and| $3$ we |
|000023b0| 68 61 76 65 20 0d 0a 5c | 62 65 67 69 6e 7b 65 71 |have ..\|begin{eq|
|000023c0| 6e 61 72 72 61 79 2a 7d | 0d 0a 31 5c 77 65 64 67 |narray*}|..1\wedg|
|000023d0| 65 20 5c 6c 65 66 74 28 | 20 32 5c 76 65 65 20 33 |e \left(| 2\vee 3|
|000023e0| 5c 72 69 67 68 74 29 20 | 26 3d 26 5c 61 6c 6c 6f |\right) |&=&\allo|
|000023f0| 77 62 72 65 61 6b 20 31 | 5c 71 71 75 61 64 20 5c |wbreak 1|\qquad \|
|00002400| 74 65 78 74 7b 61 6e 64 | 7d 5c 71 71 75 61 64 20 |text{and|}\qquad |
|00002410| 5c 6c 65 66 74 28 0d 0a | 31 5c 77 65 64 67 65 20 |\left(..|1\wedge |
|00002420| 32 5c 72 69 67 68 74 29 | 20 5c 76 65 65 20 5c 6c |2\right)| \vee \l|
|00002430| 65 66 74 28 20 31 5c 77 | 65 64 67 65 20 33 5c 72 |eft( 1\w|edge 3\r|
|00002440| 69 67 68 74 29 20 3d 31 | 20 5c 5c 0d 0a 32 5c 77 |ight) =1| \\..2\w|
|00002450| 65 64 67 65 20 5c 6c 65 | 66 74 28 20 33 5c 76 65 |edge \le|ft( 3\ve|
|00002460| 65 20 31 5c 72 69 67 68 | 74 29 20 26 3d 26 5c 61 |e 1\righ|t) &=&\a|
|00002470| 6c 6c 6f 77 62 72 65 61 | 6b 20 32 5c 71 71 75 61 |llowbrea|k 2\qqua|
|00002480| 64 20 5c 74 65 78 74 7b | 61 6e 64 7d 5c 71 71 75 |d \text{|and}\qqu|
|00002490| 61 64 20 5c 6c 65 66 74 | 28 0d 0a 32 5c 77 65 64 |ad \left|(..2\wed|
|000024a0| 67 65 20 33 5c 72 69 67 | 68 74 29 20 5c 76 65 65 |ge 3\rig|ht) \vee|
|000024b0| 20 5c 6c 65 66 74 28 20 | 32 5c 77 65 64 67 65 20 | \left( |2\wedge |
|000024c0| 31 5c 72 69 67 68 74 29 | 20 3d 32 20 5c 5c 0d 0a |1\right)| =2 \\..|
|000024d0| 33 5c 77 65 64 67 65 20 | 5c 6c 65 66 74 28 20 31 |3\wedge |\left( 1|
|000024e0| 5c 76 65 65 20 32 5c 72 | 69 67 68 74 29 20 26 3d |\vee 2\r|ight) &=|
|000024f0| 26 5c 61 6c 6c 6f 77 62 | 72 65 61 6b 20 32 5c 71 |&\allowb|reak 2\q|
|00002500| 71 75 61 64 20 5c 74 65 | 78 74 7b 61 6e 64 7d 5c |quad \te|xt{and}\|
|00002510| 71 71 75 61 64 20 5c 6c | 65 66 74 28 0d 0a 33 5c |qquad \l|eft(..3\|
|00002520| 77 65 64 67 65 20 31 5c | 72 69 67 68 74 29 20 5c |wedge 1\|right) \|
|00002530| 76 65 65 20 5c 6c 65 66 | 74 28 20 33 5c 77 65 64 |vee \lef|t( 3\wed|
|00002540| 67 65 20 32 5c 72 69 67 | 68 74 29 20 3d 32 0d 0a |ge 2\rig|ht) =2..|
|00002550| 5c 65 6e 64 7b 65 71 6e | 61 72 72 61 79 2a 7d 0d |\end{eqn|array*}.|
|00002560| 0a 53 69 6d 69 6c 61 72 | 6c 79 2c 20 0d 0a 5c 62 |.Similar|ly, ..\b|
|00002570| 65 67 69 6e 7b 65 71 6e | 61 72 72 61 79 2a 7d 0d |egin{eqn|array*}.|
|00002580| 0a 31 5c 76 65 65 20 5c | 6c 65 66 74 28 20 32 5c |.1\vee \|left( 2\|
|00002590| 77 65 64 67 65 20 33 5c | 72 69 67 68 74 29 20 26 |wedge 3\|right) &|
|000025a0| 3d 26 5c 61 6c 6c 6f 77 | 62 72 65 61 6b 20 32 5c |=&\allow|break 2\|
|000025b0| 71 71 75 61 64 20 5c 74 | 65 78 74 7b 61 6e 64 7d |qquad \t|ext{and}|
|000025c0| 5c 71 71 75 61 64 20 5c | 6c 65 66 74 28 0d 0a 31 |\qquad \|left(..1|
|000025d0| 5c 76 65 65 20 32 5c 72 | 69 67 68 74 29 20 5c 77 |\vee 2\r|ight) \w|
|000025e0| 65 64 67 65 20 5c 6c 65 | 66 74 28 20 31 5c 76 65 |edge \le|ft( 1\ve|
|000025f0| 65 20 33 5c 72 69 67 68 | 74 29 20 3d 32 20 5c 5c |e 3\righ|t) =2 \\|
|00002600| 0d 0a 32 5c 76 65 65 20 | 5c 6c 65 66 74 28 20 31 |..2\vee |\left( 1|
|00002610| 5c 77 65 64 67 65 20 33 | 5c 72 69 67 68 74 29 20 |\wedge 3|\right) |
|00002620| 26 3d 26 5c 61 6c 6c 6f | 77 62 72 65 61 6b 20 32 |&=&\allo|wbreak 2|
|00002630| 5c 71 71 75 61 64 20 5c | 74 65 78 74 7b 61 6e 64 |\qquad \|text{and|
|00002640| 7d 5c 71 71 75 61 64 20 | 5c 6c 65 66 74 28 0d 0a |}\qquad |\left(..|
|00002650| 32 5c 76 65 65 20 31 5c | 72 69 67 68 74 29 20 5c |2\vee 1\|right) \|
|00002660| 77 65 64 67 65 20 5c 6c | 65 66 74 28 20 32 5c 76 |wedge \l|eft( 2\v|
|00002670| 65 65 20 33 5c 72 69 67 | 68 74 29 20 3d 32 20 5c |ee 3\rig|ht) =2 \|
|00002680| 5c 0d 0a 33 5c 76 65 65 | 20 5c 6c 65 66 74 28 20 |\..3\vee| \left( |
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