New Year's Day Puzzle Question...

1. Dec 31, 2015

Cosmos

Let 0 ≤a≤b and a,b≠0 and a≠b
such that (ab) divided by (a+b)=2016

2. Dec 31, 2015

mathman

$b=\frac{2016a}{a-2016}$. Any a, 4032>a>2016 will do.

3. Jan 1, 2016

Cosmos

what if a and b are necessarily integers.....i tried it took too long.....

4. Jan 1, 2016

collinsmark

[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 $\neq$ b.]

Last edited: Jan 1, 2016
5. Jan 1, 2016

collinsmark

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 $\neq$ -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

6. Jan 2, 2016

Cosmos

how did you solve that......ANY TRICK ????? OR just "hit and try''

7. Jan 2, 2016

collinsmark

I wrote a computer program.

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