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Two Rectangles (Posted on 2013-12-09) Difficulty: 2 of 5
Find two rectangles, with integral sides, such that the area of the first is three times the area of the second, and the perimeter of the second is three times the perimeter of the first.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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Solution computer solutions Comment 2 of 2 |

DEFDBL A-Z
CLS
FOR t = 2 TO 9999
 FOR a = 1 TO t / 2
  b = t - a
  area1 = a * b
  IF area1 MOD 3 = 0 AND gcd(a, b) = 1 THEN
    speri1 = t
    speri2 = t * 3
    FOR a2 = 1 TO speri2 / 2
      b2 = speri2 - a2
      IF area1 = 3 * a2 * b2 THEN
        PRINT a; b, a2; b2
        ct = ct + 1
        IF ct MOD 44 = 0 THEN
           DO: LOOP UNTIL INKEY$ > ""
           PRINT
        END IF
      END IF
    NEXT
  END IF
 NEXT
NEXT t

FUNCTION gcd (a, b)
 x = a: y = b
 DO
  q = INT(x / y)
  r = x - y * q
  x = y: y = r
 LOOP UNTIL r = 0
 gcd = x
END FUNCTION


keeps on finding solutions. The first two pages are below. Multiples of the shown solutions are not shown, but there would be infinitely many for each solution shown, such as 12x15,1x110 gives rise to 24x30, 2x220, etc., as areas of both rectangles increase by the same square factor and perimeters by the same linear factor.

1st rect     2nd rect.
15  22        1  110
12  35        1  140
11  48        1  176
31  42        2  217
26  57        2  247
10  87        1  290
53  54        3  318
40  81        3  360
21  122       2  427
68  75        4  425
84  95        5  532
49  132       4  539
71  120       5  568
70  123       5  574
81  158       6  711
58  195       5  754
112  141      7  752
102  161      7  782
89  210       7  890
111  200      8  925
67  270       6  1005
19  330       2  1045
165  194      10  1067
177  220      11  1180
29  378       3  1218
129  290      10  1247
51  370       5  1258
107  324      9  1284
76  357       7  1292
62  405       6  1395
138  341      11  1426
39  452       4  1469
230  273      14  1495
115  402      10  1541
114  415      10  1577
85  456       8  1615
125  462      11  1750
140  459      12  1785
175  438      14  1825
189  460      15  1932
135  524      12  1965
94  567       9  1974
185  486      15  1998
266  405      18  1995
231  442      17  2002
165  518      14  2035
48  695       5  2224
28  729       3  2268
69  700       7  2300
249  532      19  2324
276  505      20  2323
103  690      10  2369
287  540      21  2460
319  510      22  2465
183  656      16  2501
255  596      20  2533
302  561      22  2567
344  567      24  2709
150  763      14  2725
192  731      17  2752
112  825      11  2800
162  785      15  2826
161  810      15  2898
160  837      15  2976
285  736      23  3040
47  1020      5  3196
399  668      28  3173
243  826      21  3186
486  595      30  3213
470  621      30  3243
121  972      12  3267
210  893      19  3290
238  891      21  3366
303  850      25  3434
98  1065      10  3479
539  648      33  3528
179  1020     17  3580
219  980      20  3577
595  618      34  3605
348  889      28  3683
609  640      35  3712
567  740      36  3885
374  945      30  3927
37  1284      4  3959
654  703      38  4033
212  1155     20  4081
66  1337      7  4202
237  1166     22  4187

  Posted by Charlie on 2013-12-09 13:28:31
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