The least common multiple (LCM) of four positive integers p, q, r, and s is equal to p+q+r+s
Is p*q*r*s always divisible by at least one of 3 and 5?
- If so, prove it.
- If not, provide a counterexample.
While the following is not a proof - it may help lead to one.All the solutions seem to be multiples
of the first unique 9 solutions, numbered 1-7,9,10 below
A list of sums factored is here, the programs here and here.
ind (n x ind) p q r s sum
-----------------------------------------
1 ( 1x # 1) 1 1 4 6 12
2 ( 1x # 2) 1 2 2 5 10
3 ( 1x # 3) 1 2 6 9 18
4 ( 1x # 4) 1 3 4 4 12
5 ( 1x # 5) 1 3 8 12 24
6 ( 1x # 6) 1 4 5 10 20
7 ( 1x # 7) 1 6 14 21 42
8 ( 2x # 1) 2 2 8 12 24
9 ( 1x # 9) 2 3 3 4 12
10 ( 1x #10) 2 3 10 15 30
11 ( 2x # 2) 2 4 4 10 20
12 ( 2x # 3) 2 4 12 18 36
13 ( 2x # 4) 2 6 8 8 24
14 ( 2x # 5) 2 6 16 24 48
15 ( 2x # 6) 2 8 10 20 40
16 ( 2x # 7) 2 12 28 42 84
17 ( 3x # 1) 3 3 12 18 36
18 ( 3x # 2) 3 6 6 15 30
19 ( 3x # 3) 3 6 18 27 54
20 ( 3x # 4) 3 9 12 12 36
21 ( 3x # 5) 3 9 24 36 72
22 ( 3x # 6) 3 12 15 30 60
23 ( 3x # 7) 3 18 42 63 126
24 ( 4x # 1) 4 4 16 24 48
25 ( 2x # 9) 4 6 6 8 24
26 ( 2x #10) 4 6 20 30 60
27 ( 4x # 2) 4 8 8 20 40
28 ( 4x # 3) 4 8 24 36 72
29 ( 4x # 4) 4 12 16 16 48
30 ( 4x # 5) 4 12 32 48 96
31 ( 4x # 6) 4 16 20 40 80
32 ( 4x # 7) 4 24 56 84 168
33 ( 5x # 1) 5 5 20 30 60
34 ( 5x # 2) 5 10 10 25 50
35 ( 5x # 3) 5 10 30 45 90
36 ( 5x # 4) 5 15 20 20 60
37 ( 5x # 5) 5 15 40 60 120
38 ( 5x # 6) 5 20 25 50 100
39 ( 5x # 7) 5 30 70 105 210
40 ( 6x # 1) 6 6 24 36 72
41 ( 3x # 9) 6 9 9 12 36
42 ( 3x #10) 6 9 30 45 90
43 ( 6x # 2) 6 12 12 30 60
44 ( 6x # 3) 6 12 36 54 108
45 ( 6x # 4) 6 18 24 24 72
46 ( 6x # 5) 6 18 48 72 144
47 ( 6x # 6) 6 24 30 60 120
48 ( 6x # 7) 6 36 84 126 252
49 ( 7x # 1) 7 7 28 42 84
50 ( 7x # 2) 7 14 14 35 70
51 ( 7x # 3) 7 14 42 63 126
52 ( 7x # 4) 7 21 28 28 84
53 ( 7x # 5) 7 21 56 84 168
54 ( 7x # 6) 7 28 35 70 140
55 ( 7x # 7) 7 42 98 147 294
56 ( 8x # 1) 8 8 32 48 96
57 ( 4x # 9) 8 12 12 16 48
58 ( 4x #10) 8 12 40 60 120
59 ( 8x # 2) 8 16 16 40 80
60 ( 8x # 3) 8 16 48 72 144
61 ( 8x # 4) 8 24 32 32 96
62 ( 8x # 5) 8 24 64 96 192
63 ( 8x # 6) 8 32 40 80 160
64 ( 8x # 7) 8 48 112 168 336
65 ( 9x # 1) 9 9 36 54 108
66 ( 9x # 2) 9 18 18 45 90
67 ( 9x # 3) 9 18 54 81 162
68 ( 9x # 4) 9 27 36 36 108
69 ( 9x # 5) 9 27 72 108 216
70 ( 9x # 6) 9 36 45 90 180
71 ( 9x # 7) 9 54 126 189 378
72 (10x # 1) 10 10 40 60 120
73 ( 5x # 9) 10 15 15 20 60
74 ( 5x #10) 10 15 50 75 150
75 (10x # 2) 10 20 20 50 100
76 (10x # 3) 10 20 60 90 180
77 (10x # 4) 10 30 40 40 120
78 (10x # 5) 10 30 80 120 240
79 (10x # 6) 10 40 50 100 200
80 (10x # 7) 10 60 140 210 420
81 (11x # 1) 11 11 44 66 132
82 (11x # 2) 11 22 22 55 110
83 (11x # 3) 11 22 66 99 198
84 (11x # 4) 11 33 44 44 132
85 (11x # 5) 11 33 88 132 264
86 (11x # 6) 11 44 55 110 220
87 (11x # 7) 11 66 154 231 462
88 (12x # 1) 12 12 48 72 144
89 ( 6x # 9) 12 18 18 24 72
90 ( 6x #10) 12 18 60 90 180
91 (12x # 2) 12 24 24 60 120
92 (12x # 3) 12 24 72 108 216
93 (12x # 4) 12 36 48 48 144
94 (12x # 5) 12 36 96 144 288
95 (12x # 6) 12 48 60 120 240
96 (12x # 7) 12 72 168 252 504
97 (13x # 1) 13 13 52 78 156
98 (13x # 2) 13 26 26 65 130
99 (13x # 3) 13 26 78 117 234
100 (13x # 4) 13 39 52 52 156
101 (13x # 5) 13 39 104 156 312
102 (13x # 6) 13 52 65 130 260
103 (13x # 7) 13 78 182 273 546
104 (14x # 1) 14 14 56 84 168
105 ( 7x # 9) 14 21 21 28 84
106 ( 7x #10) 14 21 70 105 210
107 (14x # 2) 14 28 28 70 140
108 (14x # 3) 14 28 84 126 252
109 (14x # 4) 14 42 56 56 168
110 (14x # 5) 14 42 112 168 336
111 (14x # 6) 14 56 70 140 280
112 (14x # 7) 14 84 196 294 588
113 (15x # 1) 15 15 60 90 180
114 (15x # 2) 15 30 30 75 150
115 (15x # 3) 15 30 90 135 270
116 (15x # 4) 15 45 60 60 180
117 (15x # 5) 15 45 120 180 360
118 (15x # 6) 15 60 75 150 300
119 (15x # 7) 15 90 210 315 630
120 (16x # 1) 16 16 64 96 192
121 ( 8x # 9) 16 24 24 32 96
122 ( 8x #10) 16 24 80 120 240
123 (16x # 2) 16 32 32 80 160
124 (16x # 3) 16 32 96 144 288
125 (16x # 4) 16 48 64 64 192
126 (16x # 5) 16 48 128 192 384
127 (16x # 6) 16 64 80 160 320
128 (16x # 7) 16 96 224 336 672
129 (17x # 1) 17 17 68 102 204
130 (17x # 2) 17 34 34 85 170
131 (17x # 3) 17 34 102 153 306
132 (17x # 4) 17 51 68 68 204
133 (17x # 5) 17 51 136 204 408
134 (17x # 6) 17 68 85 170 340
135 (17x # 7) 17 102 238 357 714
136 (18x # 1) 18 18 72 108 216
137 ( 9x # 9) 18 27 27 36 108
138 ( 9x #10) 18 27 90 135 270
139 (18x # 2) 18 36 36 90 180
140 (18x # 3) 18 36 108 162 324
141 (18x # 4) 18 54 72 72 216
142 (18x # 5) 18 54 144 216 432
143 (18x # 6) 18 72 90 180 360
144 (18x # 7) 18 108 252 378 756
145 (19x # 1) 19 19 76 114 228
146 (19x # 2) 19 38 38 95 190
147 (19x # 3) 19 38 114 171 342
148 (19x # 4) 19 57 76 76 228
149 (19x # 5) 19 57 152 228 456
150 (19x # 6) 19 76 95 190 380
151 (19x # 7) 19 114 266 399 798
152 (20x # 1) 20 20 80 120 240
153 (10x # 9) 20 30 30 40 120
154 (10x #10) 20 30 100 150 300
155 (20x # 2) 20 40 40 100 200
156 (20x # 3) 20 40 120 180 360
157 (20x # 4) 20 60 80 80 240
158 (20x # 5) 20 60 160 240 480
159 (20x # 6) 20 80 100 200 400
160 (20x # 7) 20 120 280 420 840
161 (21x # 1) 21 21 84 126 252
162 (21x # 2) 21 42 42 105 210
163 (21x # 3) 21 42 126 189 378
164 (21x # 4) 21 63 84 84 252
165 (21x # 5) 21 63 168 252 504
166 (21x # 6) 21 84 105 210 420
167 (21x # 7) 21 126 294 441 882
168 (22x # 1) 22 22 88 132 264
169 (11x # 9) 22 33 33 44 132
170 (11x #10) 22 33 110 165 330
171 (22x # 2) 22 44 44 110 220
172 (22x # 3) 22 44 132 198 396
173 (22x # 4) 22 66 88 88 264
174 (22x # 5) 22 66 176 264 528
175 (22x # 6) 22 88 110 220 440
176 (22x # 7) 22 132 308 462 924
177 (23x # 1) 23 23 92 138 276
178 (23x # 2) 23 46 46 115 230
179 (23x # 3) 23 46 138 207 414
180 (23x # 4) 23 69 92 92 276
181 (23x # 5) 23 69 184 276 552
182 (23x # 6) 23 92 115 230 460
183 (23x # 7) 23 138 322 483 966
184 (24x # 1) 24 24 96 144 288
185 (12x # 9) 24 36 36 48 144
186 (12x #10) 24 36 120 180 360
187 (24x # 2) 24 48 48 120 240
188 (24x # 3) 24 48 144 216 432
189 (24x # 4) 24 72 96 96 288
190 (24x # 5) 24 72 192 288 576
191 (24x # 6) 24 96 120 240 480
192 (24x # 7) 24 144 336 504 1008
193 (25x # 1) 25 25 100 150 300
194 (25x # 2) 25 50 50 125 250
195 (25x # 3) 25 50 150 225 450
196 (25x # 4) 25 75 100 100 300
197 (25x # 5) 25 75 200 300 600
198 (25x # 6) 25 100 125 250 500
199 (25x # 7) 25 150 350 525 1050
200 (26x # 1) 26 26 104 156 312
201 (13x # 9) 26 39 39 52 156
202 (13x #10) 26 39 130 195 390
203 (26x # 2) 26 52 52 130 260
204 (26x # 3) 26 52 156 234 468
205 (26x # 4) 26 78 104 104 312
206 (26x # 5) 26 78 208 312 624
207 (26x # 6) 26 104 130 260 520
208 (26x # 7) 26 156 364 546 1092
209 (27x # 1) 27 27 108 162 324
210 (27x # 2) 27 54 54 135 270
211 (27x # 3) 27 54 162 243 486
212 (27x # 4) 27 81 108 108 324
213 (27x # 5) 27 81 216 324 648
214 (27x # 6) 27 108 135 270 540
215 (27x # 7) 27 162 378 567 1134
216 (28x # 1) 28 28 112 168 336
217 (14x # 9) 28 42 42 56 168
218 (14x #10) 28 42 140 210 420
219 (28x # 2) 28 56 56 140 280
220 (28x # 3) 28 56 168 252 504
221 (28x # 4) 28 84 112 112 336
222 (28x # 5) 28 84 224 336 672
223 (28x # 6) 28 112 140 280 560
224 (28x # 7) 28 168 392 588 1176
225 (29x # 1) 29 29 116 174 348
226 (29x # 2) 29 58 58 145 290
227 (29x # 3) 29 58 174 261 522
228 (29x # 4) 29 87 116 116 348
229 (29x # 5) 29 87 232 348 696
230 (29x # 6) 29 116 145 290 580
231 (29x # 7) 29 174 406 609 1218
232 (30x # 1) 30 30 120 180 360
233 (15x # 9) 30 45 45 60 180
234 (15x #10) 30 45 150 225 450
235 (30x # 2) 30 60 60 150 300
236 (30x # 3) 30 60 180 270 540
237 (30x # 4) 30 90 120 120 360
238 (30x # 5) 30 90 240 360 720
239 (30x # 6) 30 120 150 300 600
240 (30x # 7) 30 180 420 630 1260
241 (31x # 1) 31 31 124 186 372
242 (31x # 2) 31 62 62 155 310
243 (31x # 3) 31 62 186 279 558
244 (31x # 4) 31 93 124 124 372
245 (31x # 5) 31 93 248 372 744
246 (31x # 6) 31 124 155 310 620
247 (31x # 7) 31 186 434 651 1302
248 (32x # 1) 32 32 128 192 384
249 (16x # 9) 32 48 48 64 192
250 (16x #10) 32 48 160 240 480
251 (32x # 2) 32 64 64 160 320
252 (32x # 3) 32 64 192 288 576
253 (32x # 4) 32 96 128 128 384
254 (32x # 5) 32 96 256 384 768
255 (32x # 6) 32 128 160 320 640
256 (32x # 7) 32 192 448 672 1344
257 (33x # 1) 33 33 132 198 396
258 (33x # 2) 33 66 66 165 330
259 (33x # 3) 33 66 198 297 594
260 (33x # 4) 33 99 132 132 396
261 (33x # 5) 33 99 264 396 792
262 (33x # 6) 33 132 165 330 660
263 (33x # 7) 33 198 462 693 1386
264 (34x # 1) 34 34 136 204 408
265 (17x # 9) 34 51 51 68 204
266 (17x #10) 34 51 170 255 510
267 (34x # 2) 34 68 68 170 340
268 (34x # 3) 34 68 204 306 612
269 (34x # 4) 34 102 136 136 408
270 (34x # 5) 34 102 272 408 816
271 (34x # 6) 34 136 170 340 680
272 (34x # 7) 34 204 476 714 1428
273 (35x # 1) 35 35 140 210 420
274 (35x # 2) 35 70 70 175 350
275 (35x # 3) 35 70 210 315 630
276 (35x # 4) 35 105 140 140 420
277 (35x # 5) 35 105 280 420 840
278 (35x # 6) 35 140 175 350 700
279 (35x # 7) 35 210 490 735 1470
280 (36x # 1) 36 36 144 216 432
281 (18x # 9) 36 54 54 72 216
282 (18x #10) 36 54 180 270 540
283 (36x # 2) 36 72 72 180 360
284 (36x # 3) 36 72 216 324 648
285 (36x # 4) 36 108 144 144 432
286 (36x # 5) 36 108 288 432 864
287 (36x # 6) 36 144 180 360 720
288 (36x # 7) 36 216 504 756 1512
289 (37x # 1) 37 37 148 222 444
290 (37x # 2) 37 74 74 185 370
291 (37x # 3) 37 74 222 333 666
292 (37x # 4) 37 111 148 148 444
293 (37x # 5) 37 111 296 444 888
294 (37x # 6) 37 148 185 370 740
295 (37x # 7) 37 222 518 777 1554
296 (38x # 1) 38 38 152 228 456
297 (19x # 9) 38 57 57 76 228
298 (19x #10) 38 57 190 285 570
299 (38x # 2) 38 76 76 190 380
300 (38x # 3) 38 76 228 342 684
301 (38x # 4) 38 114 152 152 456
302 (38x # 5) 38 114 304 456 912
303 (38x # 6) 38 152 190 380 760
304 (38x # 7) 38 228 532 798 1596
305 (39x # 1) 39 39 156 234 468
306 (39x # 2) 39 78 78 195 390
307 (39x # 3) 39 78 234 351 702
308 (39x # 4) 39 117 156 156 468
309 (39x # 5) 39 117 312 468 936
310 (39x # 6) 39 156 195 390 780
311 (39x # 7) 39 234 546 819 1638
312 (40x # 1) 40 40 160 240 480
313 (20x # 9) 40 60 60 80 240
314 (20x #10) 40 60 200 300 600
315 (40x # 2) 40 80 80 200 400
316 (40x # 3) 40 80 240 360 720
317 (40x # 4) 40 120 160 160 480
318 (40x # 5) 40 120 320 480 960
319 (40x # 6) 40 160 200 400 800
320 (40x # 7) 40 240 560 840 1680
321 (41x # 1) 41 41 164 246 492
322 (41x # 2) 41 82 82 205 410
323 (41x # 3) 41 82 246 369 738
324 (41x # 4) 41 123 164 164 492
325 (41x # 5) 41 123 328 492 984
326 (41x # 6) 41 164 205 410 820
327 (41x # 7) 41 246 574 861 1722
328 (42x # 1) 42 42 168 252 504
329 (21x # 9) 42 63 63 84 252
330 (21x #10) 42 63 210 315 630
331 (42x # 2) 42 84 84 210 420
332 (42x # 3) 42 84 252 378 756
333 (42x # 4) 42 126 168 168 504
334 (42x # 5) 42 126 336 504 1008
335 (42x # 6) 42 168 210 420 840
336 (42x # 7) 42 252 588 882 1764
337 (43x # 1) 43 43 172 258 516
338 (43x # 2) 43 86 86 215 430
339 (43x # 3) 43 86 258 387 774
340 (43x # 4) 43 129 172 172 516
341 (43x # 5) 43 129 344 516 1032
342 (43x # 6) 43 172 215 430 860
343 (43x # 7) 43 258 602 903 1806
344 (44x # 1) 44 44 176 264 528
345 (22x # 9) 44 66 66 88 264
346 (22x #10) 44 66 220 330 660
347 (44x # 2) 44 88 88 220 440
348 (44x # 3) 44 88 264 396 792
349 (44x # 4) 44 132 176 176 528
350 (44x # 5) 44 132 352 528 1056
351 (44x # 6) 44 176 220 440 880
352 (44x # 7) 44 264 616 924 1848
353 (45x # 1) 45 45 180 270 540
354 (45x # 2) 45 90 90 225 450
355 (45x # 3) 45 90 270 405 810
356 (45x # 4) 45 135 180 180 540
357 (45x # 5) 45 135 360 540 1080
358 (45x # 6) 45 180 225 450 900
359 (45x # 7) 45 270 630 945 1890
360 (46x # 1) 46 46 184 276 552
361 (23x # 9) 46 69 69 92 276
362 (23x #10) 46 69 230 345 690
363 (46x # 2) 46 92 92 230 460
364 (46x # 3) 46 92 276 414 828
365 (46x # 4) 46 138 184 184 552
366 (46x # 5) 46 138 368 552 1104
367 (46x # 6) 46 184 230 460 920
368 (46x # 7) 46 276 644 966 1932
369 (47x # 1) 47 47 188 282 564
370 (47x # 2) 47 94 94 235 470
371 (47x # 3) 47 94 282 423 846
372 (47x # 4) 47 141 188 188 564
373 (47x # 5) 47 141 376 564 1128
374 (47x # 6) 47 188 235 470 940
375 (47x # 7) 47 282 658 987 1974
376 (48x # 1) 48 48 192 288 576
377 (24x # 9) 48 72 72 96 288
378 (24x #10) 48 72 240 360 720
379 (48x # 2) 48 96 96 240 480
380 (48x # 3) 48 96 288 432 864
381 (48x # 4) 48 144 192 192 576
382 (48x # 5) 48 144 384 576 1152
383 (48x # 6) 48 192 240 480 960
384 (49x # 1) 49 49 196 294 588
385 (49x # 2) 49 98 98 245 490
386 (49x # 3) 49 98 294 441 882
387 (49x # 4) 49 147 196 196 588
388 (49x # 5) 49 147 392 588 1176
389 (49x # 6) 49 196 245 490 980
390 (50x # 1) 50 50 200 300 600
391 (25x # 9) 50 75 75 100 300
392 (25x #10) 50 75 250 375 750
393 (50x # 2) 50 100 100 250 500
394 (50x # 3) 50 100 300 450 900
395 (50x # 4) 50 150 200 200 600
396 (50x # 5) 50 150 400 600 1200
397 (50x # 6) 50 200 250 500 1000
398 (51x # 1) 51 51 204 306 612
399 (51x # 2) 51 102 102 255 510
400 (51x # 3) 51 102 306 459 918
401 (51x # 4) 51 153 204 204 612
402 (51x # 5) 51 153 408 612 1224
403 (51x # 6) 51 204 255 510 1020
404 (52x # 1) 52 52 208 312 624
405 (26x # 9) 52 78 78 104 312
406 (26x #10) 52 78 260 390 780
407 (52x # 2) 52 104 104 260 520
408 (52x # 3) 52 104 312 468 936
409 (52x # 4) 52 156 208 208 624
410 (52x # 5) 52 156 416 624 1248
411 (52x # 6) 52 208 260 520 1040
412 (53x # 1) 53 53 212 318 636
413 (53x # 2) 53 106 106 265 530
414 (53x # 3) 53 106 318 477 954
415 (53x # 4) 53 159 212 212 636
416 (53x # 5) 53 159 424 636 1272
417 (53x # 6) 53 212 265 530 1060
418 (54x # 1) 54 54 216 324 648
419 (27x # 9) 54 81 81 108 324
420 (27x #10) 54 81 270 405 810
421 (54x # 2) 54 108 108 270 540
422 (54x # 3) 54 108 324 486 972
423 (54x # 4) 54 162 216 216 648
424 (54x # 5) 54 162 432 648 1296
425 (54x # 6) 54 216 270 540 1080
426 (55x # 1) 55 55 220 330 660
427 (55x # 2) 55 110 110 275 550
428 (55x # 3) 55 110 330 495 990
429 (55x # 4) 55 165 220 220 660
430 (55x # 5) 55 165 440 660 1320
431 (55x # 6) 55 220 275 550 1100
432 (56x # 1) 56 56 224 336 672
433 (28x # 9) 56 84 84 112 336
434 (28x #10) 56 84 280 420 840
435 (56x # 2) 56 112 112 280 560
436 (56x # 3) 56 112 336 504 1008
437 (56x # 4) 56 168 224 224 672
438 (56x # 5) 56 168 448 672 1344
439 (56x # 6) 56 224 280 560 1120
440 (57x # 1) 57 57 228 342 684
441 (57x # 2) 57 114 114 285 570
442 (57x # 3) 57 114 342 513 1026
443 (57x # 4) 57 171 228 228 684
444 (57x # 5) 57 171 456 684 1368
445 (57x # 6) 57 228 285 570 1140
446 (58x # 1) 58 58 232 348 696
447 (29x # 9) 58 87 87 116 348
448 (29x #10) 58 87 290 435 870
449 (58x # 2) 58 116 116 290 580
450 (58x # 3) 58 116 348 522 1044
451 (58x # 4) 58 174 232 232 696
452 (58x # 5) 58 174 464 696 1392
453 (58x # 6) 58 232 290 580 1160
454 (59x # 1) 59 59 236 354 708
455 (59x # 2) 59 118 118 295 590
456 (59x # 3) 59 118 354 531 1062
457 (59x # 4) 59 177 236 236 708
458 (59x # 5) 59 177 472 708 1416
459 (59x # 6) 59 236 295 590 1180
460 (60x # 1) 60 60 240 360 720
461 (30x # 9) 60 90 90 120 360
462 (30x #10) 60 90 300 450 900
463 (60x # 2) 60 120 120 300 600
464 (60x # 3) 60 120 360 540 1080
465 (60x # 4) 60 180 240 240 720
466 (60x # 5) 60 180 480 720 1440
Edited on February 1, 2023, 10:54 pm
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