Yes, some duplication will result. The method is exhaustive.
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.
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.
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.
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
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.
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
Interesting that the AB column has a distinct pattern but the CD column doesn’t show one until after 2550.
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 71113=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:
If a number passes both the 4 and 19 tests, it is divisible by 76.
That reminds me of the Cecil quote about blue moons, from this column:
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.
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!]
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.
I read about it in the Wikipedia article 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.