# Finding Numbers Divisible By 13 and 76. **URL:** \ **Category:** Factual Questions\ **Created:** [April 20, 2020, 10:32am UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176 "2020-04-20T10:32:43Z")\ **Posts on this page:** 15\ **Page:** 2
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [April 20, 2020, 5:52pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/21 "2020-04-20T17:52:34Z")
> [@FastDan1](#): > > Not sure what you are getting at here. You are duplicating results. 13_1_76_2 = 13_2_76_1 = 13_76_2 Yes, some duplication will result. The method is exhaustive. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [April 20, 2020, 6:38pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/22 "2020-04-20T18:38:53Z")
> [@Great\_Antibob](#): > > I don’t know if it’s still taught in primary school, but we’re essentially discussing the concept of a [Least Common Multiple](https://en.wikipedia.org/wiki/Least_common_multiple). The Firebug finished up elementary school less than two years ago, so I can attest that it was still being taught in Maryland elementary schools that recently. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [April 20, 2020, 6:45pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/23 "2020-04-20T18:45:45Z")
> [@Exapno\_Mapcase](#): > > Yes, some duplication will result. The method is exhaustive. So was my method. The only thing you are adding is duplication. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [April 20, 2020, 7:01pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/24 "2020-04-20T19:01:41Z")
> [@naita](#): > > So was my method. The only thing you are adding is duplication. OK, I see what you’re saying. Yes, the two methods are equivalent. ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [April 20, 2020, 8:22pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/25 "2020-04-20T20:22:38Z")
To test if a number is divisible by 13, multiple the ones digit by four and add the other digits. If the result is divisible by 13, the original number is, too. 273 --\> (3 \* 4 ) + 27 = 12 + 27 = 39. 221 --\> (1 \* 4 ) + 22 = 4 + 22 = 26 546 --\> (6 \* 4) + 54 = 24 + 54 = 78. 78 --\> ( 8 \* 4) + 7 = 32 + 7 = 39 To see if a number is divisible by 76, factor it and apply the tests for all of its factors. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [April 20, 2020, 8:42pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/26 "2020-04-20T20:42:50Z")
The OP’s observation that 1976 is divisble by 76 suggests another question: what 4 digit numbers are divisible by their last two digits? Some cases are trivial: any number ending in 02 is divisible by 2, any number ending in 04 is divisible by 4, similarly with 05, 20, 25, 50, and of course 01. These all follow from the fact that they are all factors of 100. And any number of the form ABAB is equal to 101\*AB so is divisible by AB. Other than these, there are 629 others, some of which I’d call semi-trivial, like AB08 is divisible by 8 whenever AB is even, and AB03 is divisible by 3 whenever AB is divisible by 3. 03: 1203 = 03 \* 401 03: 1503 = 03 \* 501 03: 1803 = 03 \* 601 03: 2103 = 03 \* 701 03: 2403 = 03 \* 801 03: 2703 = 03 \* 901 03: 3003 = 03 \* 1001 03: 3303 = 03 \* 1101 03: 3603 = 03 \* 1201 03: 3903 = 03 \* 1301 03: 4203 = 03 \* 1401 03: 4503 = 03 \* 1501 03: 4803 = 03 \* 1601 03: 5103 = 03 \* 1701 03: 5403 = 03 \* 1801 03: 5703 = 03 \* 1901 03: 6003 = 03 \* 2001 03: 6303 = 03 \* 2101 03: 6603 = 03 \* 2201 03: 6903 = 03 \* 2301 03: 7203 = 03 \* 2401 03: 7503 = 03 \* 2501 03: 7803 = 03 \* 2601 03: 8103 = 03 \* 2701 03: 8403 = 03 \* 2801 03: 8703 = 03 \* 2901 03: 9003 = 03 \* 3001 03: 9303 = 03 \* 3101 03: 9603 = 03 \* 3201 03: 9903 = 03 \* 3301 06: 1206 = 06 \* 201 06: 1506 = 06 \* 251 06: 1806 = 06 \* 301 06: 2106 = 06 \* 351 06: 2406 = 06 \* 401 06: 2706 = 06 \* 451 06: 3006 = 06 \* 501 06: 3306 = 06 \* 551 06: 3606 = 06 \* 601 06: 3906 = 06 \* 651 06: 4206 = 06 \* 701 06: 4506 = 06 \* 751 06: 4806 = 06 \* 801 06: 5106 = 06 \* 851 06: 5406 = 06 \* 901 06: 5706 = 06 \* 951 06: 6006 = 06 \* 1001 06: 6306 = 06 \* 1051 06: 6606 = 06 \* 1101 06: 6906 = 06 \* 1151 06: 7206 = 06 \* 1201 06: 7506 = 06 \* 1251 06: 7806 = 06 \* 1301 06: 8106 = 06 \* 1351 06: 8406 = 06 \* 1401 06: 8706 = 06 \* 1451 06: 9006 = 06 \* 1501 06: 9306 = 06 \* 1551 06: 9606 = 06 \* 1601 06: 9906 = 06 \* 1651 07: 1407 = 07 \* 201 07: 2107 = 07 \* 301 07: 2807 = 07 \* 401 07: 3507 = 07 \* 501 07: 4207 = 07 \* 601 07: 4907 = 07 \* 701 07: 5607 = 07 \* 801 07: 6307 = 07 \* 901 07: 7007 = 07 \* 1001 07: 7707 = 07 \* 1101 07: 8407 = 07 \* 1201 07: 9107 = 07 \* 1301 07: 9807 = 07 \* 1401 08: 1008 = 08 \* 126 08: 1208 = 08 \* 151 08: 1408 = 08 \* 176 08: 1608 = 08 \* 201 08: 1808 = 08 \* 226 08: 2008 = 08 \* 251 08: 2208 = 08 \* 276 08: 2408 = 08 \* 301 08: 2608 = 08 \* 326 08: 2808 = 08 \* 351 08: 3008 = 08 \* 376 08: 3208 = 08 \* 401 08: 3408 = 08 \* 426 08: 3608 = 08 \* 451 08: 3808 = 08 \* 476 08: 4008 = 08 \* 501 08: 4208 = 08 \* 526 08: 4408 = 08 \* 551 08: 4608 = 08 \* 576 08: 4808 = 08 \* 601 08: 5008 = 08 \* 626 08: 5208 = 08 \* 651 08: 5408 = 08 \* 676 08: 5608 = 08 \* 701 08: 5808 = 08 \* 726 08: 6008 = 08 \* 751 08: 6208 = 08 \* 776 08: 6408 = 08 \* 801 08: 6608 = 08 \* 826 08: 6808 = 08 \* 851 08: 7008 = 08 \* 876 08: 7208 = 08 \* 901 08: 7408 = 08 \* 926 08: 7608 = 08 \* 951 08: 7808 = 08 \* 976 08: 8008 = 08 \* 1001 08: 8208 = 08 \* 1026 08: 8408 = 08 \* 1051 08: 8608 = 08 \* 1076 08: 8808 = 08 \* 1101 08: 9008 = 08 \* 1126 08: 9208 = 08 \* 1151 08: 9408 = 08 \* 1176 08: 9608 = 08 \* 1201 08: 9808 = 08 \* 1226 09: 1809 = 09 \* 201 09: 2709 = 09 \* 301 09: 3609 = 09 \* 401 09: 4509 = 09 \* 501 09: 5409 = 09 \* 601 09: 6309 = 09 \* 701 09: 7209 = 09 \* 801 09: 8109 = 09 \* 901 09: 9009 = 09 \* 1001 09: 9909 = 09 \* 1101 11: 2211 = 11 \* 201 11: 3311 = 11 \* 301 11: 4411 = 11 \* 401 11: 5511 = 11 \* 501 11: 6611 = 11 \* 601 11: 7711 = 11 \* 701 11: 8811 = 11 \* 801 11: 9911 = 11 \* 901 12: 1512 = 12 \* 126 12: 1812 = 12 \* 151 12: 2112 = 12 \* 176 12: 2412 = 12 \* 201 12: 2712 = 12 \* 226 12: 3012 = 12 \* 251 12: 3312 = 12 \* 276 12: 3612 = 12 \* 301 12: 3912 = 12 \* 326 12: 4212 = 12 \* 351 12: 4512 = 12 \* 376 12: 4812 = 12 \* 401 12: 5112 = 12 \* 426 12: 5412 = 12 \* 451 12: 5712 = 12 \* 476 12: 6012 = 12 \* 501 12: 6312 = 12 \* 526 12: 6612 = 12 \* 551 12: 6912 = 12 \* 576 12: 7212 = 12 \* 601 12: 7512 = 12 \* 626 12: 7812 = 12 \* 651 12: 8112 = 12 \* 676 12: 8412 = 12 \* 701 12: 8712 = 12 \* 726 12: 9012 = 12 \* 751 12: 9312 = 12 \* 776 12: 9612 = 12 \* 801 12: 9912 = 12 \* 826 13: 2613 = 13 \* 201 13: 3913 = 13 \* 301 13: 5213 = 13 \* 401 13: 6513 = 13 \* 501 13: 7813 = 13 \* 601 13: 9113 = 13 \* 701 14: 2114 = 14 \* 151 14: 2814 = 14 \* 201 14: 3514 = 14 \* 251 14: 4214 = 14 \* 301 14: 4914 = 14 \* 351 14: 5614 = 14 \* 401 14: 6314 = 14 \* 451 14: 7014 = 14 \* 501 14: 7714 = 14 \* 551 14: 8414 = 14 \* 601 14: 9114 = 14 \* 651 14: 9814 = 14 \* 701 15: 1215 = 15 \* 81 15: 1815 = 15 \* 121 15: 2115 = 15 \* 141 15: 2415 = 15 \* 161 15: 2715 = 15 \* 181 15: 3015 = 15 \* 201 15: 3315 = 15 \* 221 15: 3615 = 15 \* 241 15: 3915 = 15 \* 261 15: 4215 = 15 \* 281 15: 4515 = 15 \* 301 15: 4815 = 15 \* 321 15: 5115 = 15 \* 341 15: 5415 = 15 \* 361 15: 5715 = 15 \* 381 15: 6015 = 15 \* 401 15: 6315 = 15 \* 421 15: 6615 = 15 \* 441 15: 6915 = 15 \* 461 15: 7215 = 15 \* 481 15: 7515 = 15 \* 501 15: 7815 = 15 \* 521 15: 8115 = 15 \* 541 15: 8415 = 15 \* 561 15: 8715 = 15 \* 581 15: 9015 = 15 \* 601 15: 9315 = 15 \* 621 15: 9615 = 15 \* 641 15: 9915 = 15 \* 661 16: 1216 = 16 \* 76 16: 2016 = 16 \* 126 16: 2416 = 16 \* 151 16: 2816 = 16 \* 176 16: 3216 = 16 \* 201 16: 3616 = 16 \* 226 16: 4016 = 16 \* 251 16: 4416 = 16 \* 276 16: 4816 = 16 \* 301 16: 5216 = 16 \* 326 16: 5616 = 16 \* 351 16: 6016 = 16 \* 376 16: 6416 = 16 \* 401 16: 6816 = 16 \* 426 16: 7216 = 16 \* 451 16: 7616 = 16 \* 476 16: 8016 = 16 \* 501 16: 8416 = 16 \* 526 16: 8816 = 16 \* 551 16: 9216 = 16 \* 576 16: 9616 = 16 \* 601 17: 3417 = 17 \* 201 17: 5117 = 17 \* 301 17: 6817 = 17 \* 401 17: 8517 = 17 \* 501 18: 2718 = 18 \* 151 18: 3618 = 18 \* 201 18: 4518 = 18 \* 251 18: 5418 = 18 \* 301 18: 6318 = 18 \* 351 18: 7218 = 18 \* 401 18: 8118 = 18 \* 451 18: 9018 = 18 \* 501 18: 9918 = 18 \* 551 19: 3819 = 19 \* 201 19: 5719 = 19 \* 301 19: 7619 = 19 \* 401 19: 9519 = 19 \* 501 21: 4221 = 21 \* 201 21: 6321 = 21 \* 301 21: 8421 = 21 \* 401 22: 1122 = 22 \* 51 22: 3322 = 22 \* 151 22: 4422 = 22 \* 201 22: 5522 = 22 \* 251 22: 6622 = 22 \* 301 22: 7722 = 22 \* 351 22: 8822 = 22 \* 401 22: 9922 = 22 \* 451 23: 4623 = 23 \* 201 23: 6923 = 23 \* 301 23: 9223 = 23 \* 401 24: 1224 = 24 \* 51 24: 1824 = 24 \* 76 24: 3024 = 24 \* 126 24: 3624 = 24 \* 151 24: 4224 = 24 \* 176 24: 4824 = 24 \* 201 24: 5424 = 24 \* 226 24: 6024 = 24 \* 251 24: 6624 = 24 \* 276 24: 7224 = 24 \* 301 24: 7824 = 24 \* 326 24: 8424 = 24 \* 351 24: 9024 = 24 \* 376 24: 9624 = 24 \* 401 26: 1326 = 26 \* 51 26: 3926 = 26 \* 151 26: 5226 = 26 \* 201 26: 6526 = 26 \* 251 26: 7826 = 26 \* 301 26: 9126 = 26 \* 351 27: 5427 = 27 \* 201 27: 8127 = 27 \* 301 28: 1428 = 28 \* 51 28: 2128 = 28 \* 76 28: 3528 = 28 \* 126 28: 4228 = 28 \* 151 28: 4928 = 28 \* 176 28: 5628 = 28 \* 201 28: 6328 = 28 \* 226 28: 7028 = 28 \* 251 28: 7728 = 28 \* 276 28: 8428 = 28 \* 301 28: 9128 = 28 \* 326 28: 9828 = 28 \* 351 29: 5829 = 29 \* 201 29: 8729 = 29 \* 301 30: 1230 = 30 \* 41 30: 1530 = 30 \* 51 30: 1830 = 30 \* 61 30: 2130 = 30 \* 71 30: 2430 = 30 \* 81 30: 2730 = 30 \* 91 30: 3330 = 30 \* 111 30: 3630 = 30 \* 121 30: 3930 = 30 \* 131 30: 4230 = 30 \* 141 30: 4530 = 30 \* 151 30: 4830 = 30 \* 161 30: 5130 = 30 \* 171 30: 5430 = 30 \* 181 30: 5730 = 30 \* 191 30: 6030 = 30 \* 201 30: 6330 = 30 \* 211 30: 6630 = 30 \* 221 30: 6930 = 30 \* 231 30: 7230 = 30 \* 241 30: 7530 = 30 \* 251 30: 7830 = 30 \* 261 30: 8130 = 30 \* 271 30: 8430 = 30 \* 281 30: 8730 = 30 \* 291 30: 9030 = 30 \* 301 30: 9330 = 30 \* 311 30: 9630 = 30 \* 321 30: 9930 = 30 \* 331 31: 6231 = 31 \* 201 31: 9331 = 31 \* 301 32: 1632 = 32 \* 51 32: 2432 = 32 \* 76 32: 4032 = 32 \* 126 32: 4832 = 32 \* 151 32: 5632 = 32 \* 176 32: 6432 = 32 \* 201 32: 7232 = 32 \* 226 32: 8032 = 32 \* 251 32: 8832 = 32 \* 276 32: 9632 = 32 \* 301 33: 6633 = 33 \* 201 33: 9933 = 33 \* 301 34: 1734 = 34 \* 51 34: 5134 = 34 \* 151 34: 6834 = 34 \* 201 34: 8534 = 34 \* 251 35: 1435 = 35 \* 41 35: 2135 = 35 \* 61 35: 2835 = 35 \* 81 35: 4235 = 35 \* 121 35: 4935 = 35 \* 141 35: 5635 = 35 \* 161 35: 6335 = 35 \* 181 35: 7035 = 35 \* 201 35: 7735 = 35 \* 221 35: 8435 = 35 \* 241 35: 9135 = 35 \* 261 35: 9835 = 35 \* 281 36: 1836 = 36 \* 51 36: 2736 = 36 \* 76 36: 4536 = 36 \* 126 36: 5436 = 36 \* 151 36: 6336 = 36 \* 176 36: 7236 = 36 \* 201 36: 8136 = 36 \* 226 36: 9036 = 36 \* 251 36: 9936 = 36 \* 276 37: 7437 = 37 \* 201 38: 1938 = 38 \* 51 38: 5738 = 38 \* 151 38: 7638 = 38 \* 201 38: 9538 = 38 \* 251 39: 7839 = 39 \* 201 40: 1040 = 40 \* 26 40: 1240 = 40 \* 31 40: 1440 = 40 \* 36 40: 1640 = 40 \* 41 40: 1840 = 40 \* 46 40: 2040 = 40 \* 51 40: 2240 = 40 \* 56 40: 2440 = 40 \* 61 40: 2640 = 40 \* 66 40: 2840 = 40 \* 71 40: 3040 = 40 \* 76 40: 3240 = 40 \* 81 40: 3440 = 40 \* 86 40: 3640 = 40 \* 91 40: 3840 = 40 \* 96 40: 4240 = 40 \* 106 40: 4440 = 40 \* 111 40: 4640 = 40 \* 116 40: 4840 = 40 \* 121 40: 5040 = 40 \* 126 40: 5240 = 40 \* 131 40: 5440 = 40 \* 136 40: 5640 = 40 \* 141 40: 5840 = 40 \* 146 40: 6040 = 40 \* 151 40: 6240 = 40 \* 156 40: 6440 = 40 \* 161 40: 6640 = 40 \* 166 40: 6840 = 40 \* 171 40: 7040 = 40 \* 176 40: 7240 = 40 \* 181 40: 7440 = 40 \* 186 40: 7640 = 40 \* 191 40: 7840 = 40 \* 196 40: 8040 = 40 \* 201 40: 8240 = 40 \* 206 40: 8440 = 40 \* 211 40: 8640 = 40 \* 216 40: 8840 = 40 \* 221 40: 9040 = 40 \* 226 40: 9240 = 40 \* 231 40: 9440 = 40 \* 236 40: 9640 = 40 \* 241 40: 9840 = 40 \* 246 41: 8241 = 41 \* 201 42: 2142 = 42 \* 51 42: 6342 = 42 \* 151 42: 8442 = 42 \* 201 43: 8643 = 43 \* 201 44: 1144 = 44 \* 26 44: 2244 = 44 \* 51 44: 3344 = 44 \* 76 44: 5544 = 44 \* 126 44: 6644 = 44 \* 151 44: 7744 = 44 \* 176 44: 8844 = 44 \* 201 44: 9944 = 44 \* 226 45: 1845 = 45 \* 41 45: 2745 = 45 \* 61 45: 3645 = 45 \* 81 45: 5445 = 45 \* 121 45: 6345 = 45 \* 141 45: 7245 = 45 \* 161 45: 8145 = 45 \* 181 45: 9045 = 45 \* 201 45: 9945 = 45 \* 221 46: 2346 = 46 \* 51 46: 6946 = 46 \* 151 46: 9246 = 46 \* 201 47: 9447 = 47 \* 201 48: 1248 = 48 \* 26 48: 2448 = 48 \* 51 48: 3648 = 48 \* 76 48: 6048 = 48 \* 126 48: 7248 = 48 \* 151 48: 8448 = 48 \* 176 48: 9648 = 48 \* 201 49: 9849 = 49 \* 201 52: 1352 = 52 \* 26 52: 2652 = 52 \* 51 52: 3952 = 52 \* 76 52: 6552 = 52 \* 126 52: 7852 = 52 \* 151 52: 9152 = 52 \* 176 54: 2754 = 54 \* 51 54: 8154 = 54 \* 151 55: 1155 = 55 \* 21 55: 2255 = 55 \* 41 55: 3355 = 55 \* 61 55: 4455 = 55 \* 81 55: 6655 = 55 \* 121 55: 7755 = 55 \* 141 55: 8855 = 55 \* 161 55: 9955 = 55 \* 181 56: 1456 = 56 \* 26 56: 2856 = 56 \* 51 56: 4256 = 56 \* 76 56: 7056 = 56 \* 126 56: 8456 = 56 \* 151 56: 9856 = 56 \* 176 58: 2958 = 58 \* 51 58: 8758 = 58 \* 151 60: 1260 = 60 \* 21 60: 1560 = 60 \* 26 60: 1860 = 60 \* 31 60: 2160 = 60 \* 36 60: 2460 = 60 \* 41 60: 2760 = 60 \* 46 60: 3060 = 60 \* 51 60: 3360 = 60 \* 56 60: 3660 = 60 \* 61 60: 3960 = 60 \* 66 60: 4260 = 60 \* 71 60: 4560 = 60 \* 76 60: 4860 = 60 \* 81 60: 5160 = 60 \* 86 60: 5460 = 60 \* 91 60: 5760 = 60 \* 96 60: 6360 = 60 \* 106 60: 6660 = 60 \* 111 60: 6960 = 60 \* 116 60: 7260 = 60 \* 121 60: 7560 = 60 \* 126 60: 7860 = 60 \* 131 60: 8160 = 60 \* 136 60: 8460 = 60 \* 141 60: 8760 = 60 \* 146 60: 9060 = 60 \* 151 60: 9360 = 60 \* 156 60: 9660 = 60 \* 161 60: 9960 = 60 \* 166 62: 3162 = 62 \* 51 62: 9362 = 62 \* 151 64: 1664 = 64 \* 26 64: 3264 = 64 \* 51 64: 4864 = 64 \* 76 64: 8064 = 64 \* 126 64: 9664 = 64 \* 151 65: 1365 = 65 \* 21 65: 2665 = 65 \* 41 65: 3965 = 65 \* 61 65: 5265 = 65 \* 81 65: 7865 = 65 \* 121 65: 9165 = 65 \* 141 66: 3366 = 66 \* 51 66: 9966 = 66 \* 151 68: 1768 = 68 \* 26 68: 3468 = 68 \* 51 68: 5168 = 68 \* 76 68: 8568 = 68 \* 126 70: 1470 = 70 \* 21 70: 2170 = 70 \* 31 70: 2870 = 70 \* 41 70: 3570 = 70 \* 51 70: 4270 = 70 \* 61 70: 4970 = 70 \* 71 70: 5670 = 70 \* 81 70: 6370 = 70 \* 91 70: 7770 = 70 \* 111 70: 8470 = 70 \* 121 70: 9170 = 70 \* 131 70: 9870 = 70 \* 141 72: 1872 = 72 \* 26 72: 3672 = 72 \* 51 72: 5472 = 72 \* 76 72: 9072 = 72 \* 126 74: 3774 = 74 \* 51 75: 1275 = 75 \* 17 75: 1575 = 75 \* 21 75: 1875 = 75 \* 25 75: 2175 = 75 \* 29 75: 2475 = 75 \* 33 75: 2775 = 75 \* 37 75: 3075 = 75 \* 41 75: 3375 = 75 \* 45 75: 3675 = 75 \* 49 75: 3975 = 75 \* 53 75: 4275 = 75 \* 57 75: 4575 = 75 \* 61 75: 4875 = 75 \* 65 75: 5175 = 75 \* 69 75: 5475 = 75 \* 73 75: 5775 = 75 \* 77 75: 6075 = 75 \* 81 75: 6375 = 75 \* 85 75: 6675 = 75 \* 89 75: 6975 = 75 \* 93 75: 7275 = 75 \* 97 75: 7875 = 75 \* 105 75: 8175 = 75 \* 109 75: 8475 = 75 \* 113 75: 8775 = 75 \* 117 75: 9075 = 75 \* 121 75: 9375 = 75 \* 125 75: 9675 = 75 \* 129 75: 9975 = 75 \* 133 76: 1976 = 76 \* 26 76: 3876 = 76 \* 51 76: 5776 = 76 \* 76 76: 9576 = 76 \* 126 78: 3978 = 78 \* 51 80: 1280 = 80 \* 16 80: 1680 = 80 \* 21 80: 2080 = 80 \* 26 80: 2480 = 80 \* 31 80: 2880 = 80 \* 36 80: 3280 = 80 \* 41 80: 3680 = 80 \* 46 80: 4080 = 80 \* 51 80: 4480 = 80 \* 56 80: 4880 = 80 \* 61 80: 5280 = 80 \* 66 80: 5680 = 80 \* 71 80: 6080 = 80 \* 76 80: 6480 = 80 \* 81 80: 6880 = 80 \* 86 80: 7280 = 80 \* 91 80: 7680 = 80 \* 96 80: 8480 = 80 \* 106 80: 8880 = 80 \* 111 80: 9280 = 80 \* 116 80: 9680 = 80 \* 121 82: 4182 = 82 \* 51 84: 2184 = 84 \* 26 84: 4284 = 84 \* 51 84: 6384 = 84 \* 76 85: 1785 = 85 \* 21 85: 3485 = 85 \* 41 85: 5185 = 85 \* 61 85: 6885 = 85 \* 81 86: 4386 = 86 \* 51 88: 2288 = 88 \* 26 88: 4488 = 88 \* 51 88: 6688 = 88 \* 76 90: 1890 = 90 \* 21 90: 2790 = 90 \* 31 90: 3690 = 90 \* 41 90: 4590 = 90 \* 51 90: 5490 = 90 \* 61 90: 6390 = 90 \* 71 90: 7290 = 90 \* 81 90: 8190 = 90 \* 91 90: 9990 = 90 \* 111 92: 2392 = 92 \* 26 92: 4692 = 92 \* 51 92: 6992 = 92 \* 76 94: 4794 = 94 \* 51 95: 1995 = 95 \* 21 95: 3895 = 95 \* 41 95: 5795 = 95 \* 61 95: 7695 = 95 \* 81 96: 2496 = 96 \* 26 96: 4896 = 96 \* 51 96: 7296 = 96 \* 76 98: 4998 = 98 \* 51 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [April 20, 2020, 8:59pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/27 "2020-04-20T20:59:32Z")
1976 is actually more interesting because it is divisible by 19 and 76. If my back of the envelope scratchings are correct, the last year that fits that AB and CD criterion was 1995 and the next will be 2040. 2020 trivially is. 1995 is divisible by 19 and 95. 2040 is divisible by 20 and 40. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [April 20, 2020, 9:53pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/28 "2020-04-20T21:53:12Z")
Discounting numbers of the form ABAB, I count 65 cases where ABCD is divisible by both AB and CD. 1020 = 10 \* 102 = 20 \* 51 1040 = 10 \* 104 = 40 \* 26 1050 = 10 \* 105 = 50 \* 21 1122 = 11 \* 102 = 22 \* 51 1144 = 11 \* 104 = 44 \* 26 1155 = 11 \* 105 = 55 \* 21 1224 = 12 \* 102 = 24 \* 51 1248 = 12 \* 104 = 48 \* 26 1260 = 12 \* 105 = 60 \* 21 1326 = 13 \* 102 = 26 \* 51 1352 = 13 \* 104 = 52 \* 26 1365 = 13 \* 105 = 65 \* 21 1428 = 14 \* 102 = 28 \* 51 1456 = 14 \* 104 = 56 \* 26 1470 = 14 \* 105 = 70 \* 21 1530 = 15 \* 102 = 30 \* 51 1560 = 15 \* 104 = 60 \* 26 1575 = 15 \* 105 = 75 \* 21 1632 = 16 \* 102 = 32 \* 51 1664 = 16 \* 104 = 64 \* 26 1680 = 16 \* 105 = 80 \* 21 1734 = 17 \* 102 = 34 \* 51 1768 = 17 \* 104 = 68 \* 26 1785 = 17 \* 105 = 85 \* 21 1836 = 18 \* 102 = 36 \* 51 1872 = 18 \* 104 = 72 \* 26 1890 = 18 \* 105 = 90 \* 21 1938 = 19 \* 102 = 38 \* 51 1976 = 19 \* 104 = 76 \* 26 1995 = 19 \* 105 = 95 \* 21 2040 = 20 \* 102 = 40 \* 51 2080 = 20 \* 104 = 80 \* 26 2142 = 21 \* 102 = 42 \* 51 2184 = 21 \* 104 = 84 \* 26 2244 = 22 \* 102 = 44 \* 51 2288 = 22 \* 104 = 88 \* 26 2346 = 23 \* 102 = 46 \* 51 2392 = 23 \* 104 = 92 \* 26 2448 = 24 \* 102 = 48 \* 51 2496 = 24 \* 104 = 96 \* 26 2550 = 25 \* 102 = 50 \* 51 2652 = 26 \* 102 = 52 \* 51 2754 = 27 \* 102 = 54 \* 51 2856 = 28 \* 102 = 56 \* 51 2958 = 29 \* 102 = 58 \* 51 3060 = 30 \* 102 = 60 \* 51 3162 = 31 \* 102 = 62 \* 51 3264 = 32 \* 102 = 64 \* 51 3366 = 33 \* 102 = 66 \* 51 3468 = 34 \* 102 = 68 \* 51 3570 = 35 \* 102 = 70 \* 51 3672 = 36 \* 102 = 72 \* 51 3774 = 37 \* 102 = 74 \* 51 3876 = 38 \* 102 = 76 \* 51 3978 = 39 \* 102 = 78 \* 51 4080 = 40 \* 102 = 80 \* 51 4182 = 41 \* 102 = 82 \* 51 4284 = 42 \* 102 = 84 \* 51 4386 = 43 \* 102 = 86 \* 51 4488 = 44 \* 102 = 88 \* 51 4590 = 45 \* 102 = 90 \* 51 4692 = 46 \* 102 = 92 \* 51 4794 = 47 \* 102 = 94 \* 51 4896 = 48 \* 102 = 96 \* 51 4998 = 49 \* 102 = 98 \* 51 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [April 21, 2020, 2:31am UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/29 "2020-04-21T02:31:52Z")
Interesting that the AB column has a distinct pattern but the CD column doesn’t show one until after 2550. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 21, 2020, 10:18am UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/30 "2020-04-21T10:18:54Z")
A nice way to find the remainder when dividing by 13: Going from right to left, alternate first adding and then subtracting triplets of digits. For example: 37,143,536,539: 539 - 536 + 143 - 37 = 109, which has the same remainder (mod 13) as the original number: 5 Incidentally, this method works simultaneously for 7, 11, and 13 since 7_11_13=1001. For 76=4\*19: The 4 is easy: You only need check the last two digits. 19: I didn’t know a nice one off the top of my head, but here’s one I found: > **[What is the divisibility rule of 19?](https://www.quora.com/What-is-the-divisibility-rule-of-19)** > > Answer (1 of 4): divisibility test of 19 - > > Add 2 times the last digit to the remaining number. Repeat the step . If the result is divisible by 19, the original number is also divisible by 19 > > Check for 11343- > > 1 step is that add 2 times the last... If a number passes both the 4 and 19 tests, it is divisible by 76. ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [April 21, 2020, 11:26am UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/31 "2020-04-21T11:26:58Z")
> [@Schnitte](#): > > Depends on what you expect before you consider a year eventful. 1976 was a presidential election year, and Summer Olympics took place. That reminds me of the Cecil quote about blue moons, [from this column](https://www.straightdope.com/columns/read/162/is-there-ever-really-a-blue-moon/): > [@](#): > > There is a popular conviction, which the media have done much to reinforce, that a blue moon is a second full moon occurring within a calendar month. This occurs every two or three years. So now “once in a blue moon” means “not all that often, but more often than the Olympics.” ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 21, 2020, 11:41am UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/32 "2020-04-21T11:41:20Z")
Note: for proper Gematria, it is not sufficient to consider only the arithmetical properties of 1976, but also to do stuff like translate names and phrases into Greek and Hebrew and convert them into numerical values. (For English you may use A = 1, B = 2, …, or A = 101, B = 102, …; whatever.) I will leave it as an exercise. Bonus points if you throw in a magic square or something equally esoteric. ---
**Author:** ![Schnitte](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/schnitte/32/9033_2.png) [@Schnitte](https://boards.straightdope.com/u/Schnitte)\ **Post date:** [April 21, 2020, 12:17pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/33 "2020-04-21T12:17:48Z")
> [@Ludovic](#): > > That reminds me of the Cecil quote about blue moons, [from this column](https://www.straightdope.com/columns/read/162/is-there-ever-really-a-blue-moon/): Coincidentally, I read up the history of the term “blue moon” just very recently. It seems that the original meaning was slightly different: Normally, each of the four seasons of the year (astronomically defined, i.e., running from an equinox to the subsequent solstice, or a solctice to the subsequent equinox) has three full moons, but occasionally a season has four. In such a case the third of these four moons is called the “blue moon”, to indicate that its observation does not yet announce the coming end of the season. From there it got the more general meaning of “an extra full moon”, and from there it got the widespread meaning of “a second full moon within one calendar month”, and now it just has the proverbial meaning of something rare, but not super rare. Coming back to the Olympics, I’d like to add to my previous post that 1976 did not only feature Summer Olympics (in Montreal), but also Winter Olympics (in Innsbruck) - the decoupling of the Winter Olympics from the summer event did not occur until 1994. [They could be recoupled again very soon - if the Tokyo Olympics, originally scheduled for this summer but postponed until 2021 owing to the pandemic, can’t take place in 2021 either, then it’s not far-fetched that we might have both Winter and Summer Olympics in 2022!] ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 21, 2020, 1:10pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/34 "2020-04-21T13:10:59Z")
> [@Schnitte](#): > > Coincidentally, I read up the history of the term “blue moon” just very recently. It seems that the original meaning was slightly different: Normally, each of the four seasons of the year (astronomically defined, i.e., running from an equinox to the subsequent solstice, or a solctice to the subsequent equinox) has three full moons, but occasionally a season has four. In such a case the third of these four moons is called the “blue moon”, to indicate that its observation does not yet announce the coming end of the season. From there it got the more general meaning of “an extra full moon”, and from there it got the widespread meaning of “a second full moon within one calendar month”, and now it just has the proverbial meaning of something rare, but not super rare. That is not what the column claims, though. It says a “blue moon” is supposed to indicate a patent absurdity, and gives a quote in English from 1528. Where did you read about this agricultural stuff, and what does it have to do with colours? I would not trust any farmer who tries to predict the seasons using a _lunar_ calendar. Back to numerology, to compute the phases of the moon, one starts by figuring they (currently) occur on average every 29.53059 days (approximately), so that is one magic number to work with. ---
**Author:** ![Schnitte](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/schnitte/32/9033_2.png) [@Schnitte](https://boards.straightdope.com/u/Schnitte)\ **Post date:** [April 21, 2020, 3:04pm UTC](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176/35 "2020-04-21T15:04:56Z")
> [@DPRK](#): > > Where did you read about this agricultural stuff, and what does it have to do with colours? I read about it in the [Wikipedia article](https://en.wikipedia.org/wiki/Blue_moon) on the term “blue moon”. To put that story shortly, the reference to the colour blue is, according to that theory, ret-conned; it comes from a similarly sounding old word meaning betrayal (because the “blue moon”, i.e. the third full moon in a season with four, falsely announces an end to the season) and was, much later when that other word had become forgotten, linked to the colour. It’s not an uncontested etymology, but it doesn’t sound plainly implausible. It does not, by the way, require farmers to rely on a lunar calendar to predict the seasons - quite to the contrary, it shows that (assuming that the etymology is correct) farmers were very well aware that the lunar cycles are unreliable in determining the seasons of the solar year. [Previous page](https://boards.straightdope.com/t/finding-numbers-divisible-by-13-and-76/852176.md?page=1)