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New Year's Day Puzzle Question...

  1. Dec 31, 2015 #1
    Let 0 ≤a≤b and a,b≠0 and a≠b
    such that (ab) divided by (a+b)=2016
     
  2. jcsd
  3. Dec 31, 2015 #2

    mathman

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    [itex]b=\frac{2016a}{a-2016}[/itex]. Any a, 4032>a>2016 will do.
     
  4. Jan 1, 2016 #3
    what if a and b are necessarily integers.....i tried it took too long....:frown:.
     
  5. Jan 1, 2016 #4

    collinsmark

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    [Edit: for a and b being positive integers (i.e., natural numbers)]

    a = 2017, b = 4066272
    a = 2018, b = 2034144
    a = 2019, b = 1356768
    a = 2020, b = 1018080
    a = 2022, b = 679392
    a = 2023, b = 582624
    a = 2024, b = 510048
    a = 2025, b = 453600
    a = 2028, b = 340704
    a = 2030, b = 292320
    a = 2032, b = 256032
    a = 2034, b = 227808
    a = 2037, b = 195552
    a = 2040, b = 171360
    a = 2043, b = 152544
    a = 2044, b = 147168
    a = 2048, b = 129024
    a = 2052, b = 114912
    a = 2058, b = 98784
    a = 2064, b = 86688
    a = 2065, b = 84960
    a = 2070, b = 77280
    a = 2072, b = 74592
    a = 2079, b = 66528
    a = 2080, b = 65520
    a = 2088, b = 58464
    a = 2097, b = 52192
    a = 2100, b = 50400
    a = 2112, b = 44352
    a = 2114, b = 43488
    a = 2124, b = 39648
    a = 2128, b = 38304
    a = 2142, b = 34272
    a = 2144, b = 33768
    a = 2160, b = 30240
    a = 2163, b = 29664
    a = 2178, b = 27104
    a = 2184, b = 26208
    a = 2205, b = 23520
    a = 2208, b = 23184
    a = 2212, b = 22752
    a = 2232, b = 20832
    a = 2240, b = 20160
    a = 2268, b = 18144
    a = 2272, b = 17892
    a = 2304, b = 16128
    a = 2310, b = 15840
    a = 2340, b = 14560
    a = 2352, b = 14112
    a = 2394, b = 12768
    a = 2400, b = 12600
    a = 2408, b = 12384
    a = 2448, b = 11424
    a = 2457, b = 11232
    a = 2464, b = 11088
    a = 2520, b = 10080
    a = 2528, b = 9954
    a = 2583, b = 9184
    a = 2592, b = 9072
    a = 2604, b = 8928
    a = 2664, b = 8288
    a = 2688, b = 8064
    a = 2772, b = 7392
    a = 2784, b = 7308
    a = 2800, b = 7200
    a = 2880, b = 6720
    a = 2898, b = 6624
    a = 2912, b = 6552
    a = 3024, b = 6048
    a = 3040, b = 5985
    a = 3150, b = 5600
    a = 3168, b = 5544
    a = 3192, b = 5472
    a = 3312, b = 5152
    a = 3339, b = 5088
    a = 3360, b = 5040
    a = 3528, b = 4704
    a = 3552, b = 4662
    a = 3584, b = 4608
    a = 3744, b = 4368
    a = 3780, b = 4320
    a = 3808, b = 4284
    a = 4032, b = 4032 [Edit: Oops. I see now that this last entry is restricted in the problem statement, a [itex] \neq [/itex] b.]
     
    Last edited: Jan 1, 2016
  6. Jan 1, 2016 #5

    collinsmark

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    Perhaps this is more complete.

    Since the problem does not technically restrict a or b from being negative, here is a more complete list where they are allowed to be negative (such that both a and b are nonzero integers). I did have to add an additional restriction that a [itex] \neq [/itex] -b to avoid divide by zero.

    a = -4062240, b = 2015
    a = -2030112, b = 2014
    a = -1352736, b = 2013
    a = -1014048, b = 2012
    a = -675360, b = 2010
    a = -578592, b = 2009
    a = -506016, b = 2008
    a = -449568, b = 2007
    a = -336672, b = 2004
    a = -288288, b = 2002
    a = -252000, b = 2000
    a = -223776, b = 1998
    a = -191520, b = 1995
    a = -167328, b = 1992
    a = -148512, b = 1989
    a = -143136, b = 1988
    a = -124992, b = 1984
    a = -110880, b = 1980
    a = -94752, b = 1974
    a = -82656, b = 1968
    a = -80928, b = 1967
    a = -73248, b = 1962
    a = -70560, b = 1960
    a = -62496, b = 1953
    a = -61488, b = 1952
    a = -54432, b = 1944
    a = -48160, b = 1935
    a = -46368, b = 1932
    a = -40320, b = 1920
    a = -39456, b = 1918
    a = -35616, b = 1908
    a = -34272, b = 1904
    a = -30240, b = 1890
    a = -29736, b = 1888
    a = -26208, b = 1872
    a = -25632, b = 1869
    a = -23072, b = 1854
    a = -22176, b = 1848
    a = -19488, b = 1827
    a = -19152, b = 1824
    a = -18720, b = 1820
    a = -16800, b = 1800
    a = -16128, b = 1792
    a = -14112, b = 1764
    a = -13860, b = 1760
    a = -12096, b = 1728
    a = -11808, b = 1722
    a = -10528, b = 1692
    a = -10080, b = 1680
    a = -8736, b = 1638
    a = -8568, b = 1632
    a = -8352, b = 1624
    a = -7392, b = 1584
    a = -7200, b = 1575
    a = -7056, b = 1568
    a = -6048, b = 1512
    a = -5922, b = 1504
    a = -5152, b = 1449
    a = -5040, b = 1440
    a = -4896, b = 1428
    a = -4256, b = 1368
    a = -4032, b = 1344
    a = -3360, b = 1260
    a = -3276, b = 1248
    a = -3168, b = 1232
    a = -2688, b = 1152
    a = -2592, b = 1134
    a = -2520, b = 1120
    a = -2016, b = 1008
    a = -1953, b = 992
    a = -1568, b = 882
    a = -1512, b = 864
    a = -1440, b = 840
    a = -1120, b = 720
    a = -1056, b = 693
    a = -1008, b = 672
    a = -672, b = 504
    a = -630, b = 480
    a = -576, b = 448
    a = -336, b = 288
    a = -288, b = 252
    a = -252, b = 224
    a = 2017, b = 4066272
    a = 2018, b = 2034144
    a = 2019, b = 1356768
    a = 2020, b = 1018080
    a = 2022, b = 679392
    a = 2023, b = 582624
    a = 2024, b = 510048
    a = 2025, b = 453600
    a = 2028, b = 340704
    a = 2030, b = 292320
    a = 2032, b = 256032
    a = 2034, b = 227808
    a = 2037, b = 195552
    a = 2040, b = 171360
    a = 2043, b = 152544
    a = 2044, b = 147168
    a = 2048, b = 129024
    a = 2052, b = 114912
    a = 2058, b = 98784
    a = 2064, b = 86688
    a = 2065, b = 84960
    a = 2070, b = 77280
    a = 2072, b = 74592
    a = 2079, b = 66528
    a = 2080, b = 65520
    a = 2088, b = 58464
    a = 2097, b = 52192
    a = 2100, b = 50400
    a = 2112, b = 44352
    a = 2114, b = 43488
    a = 2124, b = 39648
    a = 2128, b = 38304
    a = 2142, b = 34272
    a = 2144, b = 33768
    a = 2160, b = 30240
    a = 2163, b = 29664
    a = 2178, b = 27104
    a = 2184, b = 26208
    a = 2205, b = 23520
    a = 2208, b = 23184
    a = 2212, b = 22752
    a = 2232, b = 20832
    a = 2240, b = 20160
    a = 2268, b = 18144
    a = 2272, b = 17892
    a = 2304, b = 16128
    a = 2310, b = 15840
    a = 2340, b = 14560
    a = 2352, b = 14112
    a = 2394, b = 12768
    a = 2400, b = 12600
    a = 2408, b = 12384
    a = 2448, b = 11424
    a = 2457, b = 11232
    a = 2464, b = 11088
    a = 2520, b = 10080
    a = 2528, b = 9954
    a = 2583, b = 9184
    a = 2592, b = 9072
    a = 2604, b = 8928
    a = 2664, b = 8288
    a = 2688, b = 8064
    a = 2772, b = 7392
    a = 2784, b = 7308
    a = 2800, b = 7200
    a = 2880, b = 6720
    a = 2898, b = 6624
    a = 2912, b = 6552
    a = 3024, b = 6048
    a = 3040, b = 5985
    a = 3150, b = 5600
    a = 3168, b = 5544
    a = 3192, b = 5472
    a = 3312, b = 5152
    a = 3339, b = 5088
    a = 3360, b = 5040
    a = 3528, b = 4704
    a = 3552, b = 4662
    a = 3584, b = 4608
    a = 3744, b = 4368
    a = 3780, b = 4320
    a = 3808, b = 4284
     
  7. Jan 2, 2016 #6
    how did you solve that......ANY TRICK ?????:eek: OR just "hit and try'':DD
     
  8. Jan 2, 2016 #7

    collinsmark

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    I wrote a computer program. o0)

    (I'm working on your other puzzle presently.)
     
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