Sums & Products| x+y+z=a+b+c and x*y*z=a*b*c

 

Note that

118 can be partitioned into three parts that have the same product in 4 different ways:

14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72 = 118     and
14 \times 50 \times 54 = 15 \times 40 \times 63 = 18 \times 30 \times 70 = 21 \times 25 \times 72 = 37800

It is the smallest number for which that is possible.

Find the next smallest one.
 

 
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Find 6 distinct integers so that

x \; + \; y \; + \; z \; = \; a \; + \; b \; + \; c     and
x \cdot y \cdot z \; = \; a \cdot b \cdot c

 

 
Here are some solutions:

3+4+12 = 2+8+9 = 19 ……….. 3\times 4\times 12 = 2\times 8\times 9 = 144
3+4+14 = 2+7+12 = 21 ………. 4\times 3\times 14 = 2\times 7\times 12 = 168
3+5+16 = 2+10+12 = 24 ……… 3\times 5\times 16 = 2\times 10\times 12 = 240
3+5+18 = 2+15+9 = 26 ………. 3\times 5\times 18 = 2\times 15\times 9 = 270
3+5+32 = 2+8+30 = 40 ………. 3\times 5\times 32 = 2\times 8\times 30 = 480
3+6+20 = 2+12+15 = 29 ……… 3\times 6\times 20 = 2\times 12\times 15 = 360
3+6+22 = 2+11+18 = 31 ……… 3\times 6\times 22 = 2\times 11\times 18 = 396
3+6+30 = 2+10+27 = 39 ……… 3\times 6\times 30 = 2\times 10\times 27 = 540
3+7+24 = 2+14+18 = 34 ……… 3\times 7\times 24 = 2\times 14\times 18 = 504
3+7+26 = 2+13+21 = 36 ……… 3\times 7\times 26 = 2\times 13\times 21 = 546
3+7+32 = 2+12+28 = 42 ……… 3\times 7\times 32 = 2\times 12\times 28 = 672
3+8+28 = 2+16+21 = 39 ……… 3\times 8\times 28 = 2\times 16\times 21 = 672
3+8+30 = 2+15+24 = 41 ……… 3\times 8\times 30 = 2\times 15\times 24 = 720
3+8+35 = 2+14+30 = 46 ……… 3\times 8\times 35 = 2\times 14\times 30 = 840
3+8+52 = 2+13+48 = 63 ……… 3\times 8\times 52 = 2\times 13\times 48 = 1248
3+9+32 = 2+18+24 = 44 ……… 3\times 9\times 32 = 2\times 18\times 24 = 864
3+9+34 = 2+17+27 = 46 ……… 3\times 9\times 34 = 2\times 17\times 27 = 918
3+9+50 = 2+15+45 = 62 ……… 3\times 9\times 50 = 2\times 15\times 45 = 1350
3+10+35 = 2+21+25 = 48 …….. 3\times 10\times 35 = 2\times 21\times 25 = 1050
3+10+36 = 2+20+27 = 49 …….. 3\times 10\times 36 = 2\times 20\times 27 = 1080
3+10+38 = 2+19+30 = 51 …….. 3\times 10\times 38 = 2\times 19\times 30 = 1140
3+10+42 = 2+18+35 = 55 …….. 3\times 10\times 42 = 2\times 18\times 35 = 1260
3+10+51 = 2+17+45 = 64 …….. 3\times 10\times 51 = 2\times 17\times 45 = 1530
3+10+80 = 2+16+75 = 93 …….. 3\times 10\times 80 = 2\times 16\times 75 = 2400
3+11+40 = 2+22+30 = 54 …….. 3\times 11\times 40 = 2\times 22\times 30 = 1320
3+11+42 = 2+21+33 = 56 …….. 3\times 11\times 42 = 2\times 21\times 33 = 1386
3+11+72 = 2+18+66 = 86 …….. 3\times 11\times 72 = 2\times 18\times 66 = 2376
3+12+44 = 2+24+33 = 59 …….. 3\times 12\times 44 = 2\times 24\times 33 = 1584
3+12+46 = 2+23+36 = 61 …….. 3\times 12\times 46 = 2\times 23\times 36 = 1656
3+12+56 = 2+21+48 = 71 …….. 3\times 12\times 56 = 2\times 21\times 48 = 2016
3+12+70 = 2+20+63 = 85 …….. 3\times 12\times 70 = 2\times 20\times 63 = 2520
3+13+98 = 2+21+91 = 114 ……. 3\times 13\times 98 = 2\times 21\times 91 = 3822
3+14+72 = 2+24+63 = 89 …….. 3\times 14\times 72 = 2\times 24\times 63 = 3024
3+14+92 = 2+23+84 = 109 ……. 3\times 14\times 92 = 2\times 23\times 84 = 3864

4+5+24 = 2+15+16 = 33 ……… 4\times 5\times 24 = 2\times 15\times 16 = 480
4+5+26 = 2+13+20 = 35 ……… 4\times 5\times 26 = 2\times 13\times 20 = 520
4+5+30 = 2+12+25 = 39 ……… 4\times 5\times 30 = 2\times 12\times 25 = 600
4+5+44 = 2+11+40 = 53 ……… 4\times 5\times 44 = 2\times 11\times 40 = 880
4+6+30 = 2+18+20 = 40 ……… 4\times 6\times 30 = 2\times 18\times 20 = 720
4+6+32 = 2+16+24 = 42 ……… 4\times 6\times 32 = 2\times 16\times 24 = 768
4+6+35 = 2+15+28 = 45 ……… 4\times 6\times 35 = 2\times 15\times 28 = 840
4+6+42 = 2+14+36 = 52 ……… 4\times 6\times 42 = 2\times 14\times 36 = 1008
4+6+65 = 2+13+60 = 75 ……… 4\times 6\times 65 = 2\times 13\times 60 = 1560
4+7+36 = 2+24+21 = 47 ……… 4\times 7\times 36 = 2\times 24\times 21 = 1008
4+7+38 = 2+19+28 = 49 ……… 4\times 7\times 38 = 2\times 19\times 28 = 1064
4+7+56 = 2+16+49 = 67 ……… 4\times 7\times 56 = 2\times 16\times 49 = 1568
4+7+90 = 2+15+84 = 101 …….. 4\times 7\times 90 = 2\times 15\times 84 = 2520
4+8+42 = 2+24+28 = 54 ……… 4\times 8\times 42 = 2\times 24\times 28 = 1344
4+8+44 = 2+22+32 = 56 ……… 4\times 8\times 44 = 2\times 22\times 32 = 1408
4+8+50 = 2+20+40 = 62 ……… 4\times 8\times 50 = 2\times 20\times 40 = 1600
4+8+57 = 2+19+48 = 69 ……… 4\times 8\times 57 = 2\times 19\times 48 = 1824
4+8+72 = 2+18+64 = 84 ……… 4\times 8\times 72 = 2\times 18\times 64 = 2304
4+9+52 = 2+24+39 = 65 ……… 4\times 9\times 52 = 2\times 24\times 39 = 1872
4+9+70 = 2+21+60 = 83 ……… 4\times 9\times 70 = 2\times 21\times 60 = 2520
4+9+90 = 2+20+81 = 103 …….. 4\times 9\times 90 = 2\times 20\times 81 = 3240
4+10+72 = 2+24+60 = 86 …….. 4\times 10\times 72 = 2\times 24\times 60 = 2880

5+6+40 = 2+24+25 = 51 ……… 5\times 6\times 40 = 2\times 24\times 25 = 1200
5+6+42 = 2+21+30 = 53 ……… 5\times 6\times 42 = 2\times 21\times 30 = 1260
5+6+44 = 2+20+33 = 55 ……… 5\times 6\times 44 = 2\times 20\times 33 = 1320
5+6+54 = 2+18+45 = 65 ……… 5\times 6\times 54 = 2\times 18\times 45 = 1620
5+6+68 = 2+17+60 = 79 ……… 5\times 6\times 68 = 2\times 17\times 60 = 2040
5+7+66 = 2+21+55 = 78 ……… 5\times 7\times 66 = 2\times 21\times 55 = 2310
5+7+80 = 2+20+70 = 92 ……… 5\times 7\times 80 = 2\times 20\times 70 = 2800
5+8+78 = 2+24+65 = 91 ……… 5\times 8\times 78 = 2\times 24\times 65 = 3120
5+8+92 = 2+23+80 = 105 …….. 5\times 8\times 92 = 2\times 23\times 80 = 3680

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

One Response to Sums & Products| x+y+z=a+b+c and x*y*z=a*b*c

  1. paul says:

    The next smallest one is 130. format is N {{a, b, c, product}, {….}}

    130 {{15,24,91,32760},{14,26,90,32760},{10,42,78,32760},{9,56,65,32760}}

    Paul.

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