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MathematicsMathematics
16 January, 20:32

A student at Harvard named Kevin

Was counting his stones by 11

He messed up n times

And instead counted 9s

And wound up at 2007.

To be precise, Kevin divides 2007 stones into groups of 9 stones and 11 stones, where n of the groups have 9 stones in them. How many values of could make this limerick true?

(Kevin can mess up every time)

+1
Answers (1)
  1. 16 January, 22:56
    0
    The answer is 21.

    Work:

    Some groups of 11,

    and some groups of 9,

    together make 2007.

    We may make an equation

    11x + 9y = 2007,

    and we need to find how many possibilities there are for y.

    2007/9=223

    So, the highest value for y is 223.

    Let k be a value for y.

    If k is a valid value for y then (2007-9k) must be divisible by 11.

    Let's just brute force this by doing this validity test with every number from 1 to 223. Then we can count how many valid options there are.

    0 is NOT valid

    1 is valid

    2 is NOT valid

    3 is NOT valid

    4 is NOT valid

    5 is NOT valid

    6 is NOT valid

    7 is NOT valid

    8 is NOT valid

    9 is NOT valid

    10 is NOT valid

    11 is NOT valid

    12 is valid

    13 is NOT valid

    14 is NOT valid

    15 is NOT valid

    16 is NOT valid

    17 is NOT valid

    18 is NOT valid

    19 is NOT valid

    20 is NOT valid

    21 is NOT valid

    22 is NOT valid

    23 is valid

    24 is NOT valid

    25 is NOT valid

    26 is NOT valid

    27 is NOT valid

    28 is NOT valid

    29 is NOT valid

    30 is NOT valid

    31 is NOT valid

    32 is NOT valid

    33 is NOT valid

    34 is valid

    35 is NOT valid

    36 is NOT valid

    37 is NOT valid

    38 is NOT valid

    39 is NOT valid

    40 is NOT valid

    41 is NOT valid

    42 is NOT valid

    43 is NOT valid

    44 is NOT valid

    45 is valid

    46 is NOT valid

    47 is NOT valid

    48 is NOT valid

    49 is NOT valid

    50 is NOT valid

    51 is NOT valid

    52 is NOT valid

    53 is NOT valid

    54 is NOT valid

    55 is NOT valid

    56 is valid

    57 is NOT valid

    58 is NOT valid

    59 is NOT valid

    60 is NOT valid

    61 is NOT valid

    62 is NOT valid

    63 is NOT valid

    64 is NOT valid

    65 is NOT valid

    66 is NOT valid

    67 is valid

    68 is NOT valid

    69 is NOT valid

    70 is NOT valid

    71 is NOT valid

    72 is NOT valid

    73 is NOT valid

    74 is NOT valid

    75 is NOT valid

    76 is NOT valid

    77 is NOT valid

    78 is valid

    79 is NOT valid

    80 is NOT valid

    81 is NOT valid

    82 is NOT valid

    83 is NOT valid

    84 is NOT valid

    85 is NOT valid

    86 is NOT valid

    87 is NOT valid

    88 is NOT valid

    89 is valid

    90 is NOT valid

    91 is NOT valid

    92 is NOT valid

    93 is NOT valid

    94 is NOT valid

    95 is NOT valid

    96 is NOT valid

    97 is NOT valid

    98 is NOT valid

    99 is NOT valid

    100 is valid

    101 is NOT valid

    102 is NOT valid

    103 is NOT valid

    104 is NOT valid

    105 is NOT valid

    106 is NOT valid

    107 is NOT valid

    108 is NOT valid

    109 is NOT valid

    110 is NOT valid

    111 is valid

    112 is NOT valid

    113 is NOT valid

    114 is NOT valid

    115 is NOT valid

    116 is NOT valid

    117 is NOT valid

    118 is NOT valid

    119 is NOT valid

    120 is NOT valid

    121 is NOT valid

    122 is valid

    123 is NOT valid

    124 is NOT valid

    125 is NOT valid

    126 is NOT valid

    127 is NOT valid

    128 is NOT valid

    129 is NOT valid

    130 is NOT valid

    131 is NOT valid

    132 is NOT valid

    133 is valid

    134 is NOT valid

    135 is NOT valid

    136 is NOT valid

    137 is NOT valid

    138 is NOT valid

    139 is NOT valid

    140 is NOT valid

    141 is NOT valid

    142 is NOT valid

    143 is NOT valid

    144 is valid

    145 is NOT valid

    146 is NOT valid

    147 is NOT valid

    148 is NOT valid

    149 is NOT valid

    150 is NOT valid

    151 is NOT valid

    152 is NOT valid

    153 is NOT valid

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    155 is valid

    156 is NOT valid

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    159 is NOT valid

    160 is NOT valid

    161 is NOT valid

    162 is NOT valid

    163 is NOT valid

    164 is NOT valid

    165 is NOT valid

    166 is valid

    167 is NOT valid

    168 is NOT valid

    169 is NOT valid

    170 is NOT valid

    171 is NOT valid

    172 is NOT valid

    173 is NOT valid

    174 is NOT valid

    175 is NOT valid

    176 is NOT valid

    177 is valid

    178 is NOT valid

    179 is NOT valid

    180 is NOT valid

    181 is NOT valid

    182 is NOT valid

    183 is NOT valid

    184 is NOT valid

    185 is NOT valid

    186 is NOT valid

    187 is NOT valid

    188 is valid

    189 is NOT valid

    190 is NOT valid

    191 is NOT valid

    192 is NOT valid

    193 is NOT valid

    194 is NOT valid

    195 is NOT valid

    196 is NOT valid

    197 is NOT valid

    198 is NOT valid

    199 is valid

    200 is NOT valid

    201 is NOT valid

    202 is NOT valid

    203 is NOT valid

    204 is NOT valid

    205 is NOT valid

    206 is NOT valid

    207 is NOT valid

    208 is NOT valid

    209 is NOT valid

    210 is valid

    211 is NOT valid

    212 is NOT valid

    213 is NOT valid

    214 is NOT valid

    215 is NOT valid

    216 is NOT valid

    217 is NOT valid

    218 is NOT valid

    219 is NOT valid

    220 is NOT valid

    221 is valid

    222 is NOT valid

    223 is NOT valid

    21 valid options for y!
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