## Middle digit deleted | 5-digit number

Let   $n$   be a 5-digit number:     $n = ABCDE$,   and

$m = ABDE$   a 4-digit number.

Determine all   $n$   such that   $n / m$   is an integer

math grad - Interest: Number theory
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### 4 Responses to Middle digit deleted | 5-digit number

1. paul says:

It is all the numbers

{10000, 11000….98000,99000

Paul.

2. David @InfinitelyManic says:

Partial list – having problems in printf output in terminal due to short list.

A B C D E ABCDE ABDE QUOTIENT
5 0 0 0 0 50000 5000 10
4 9 0 0 0 49000 4900 10
4 8 0 0 0 48000 4800 10
4 7 0 0 0 47000 4700 10
4 6 0 0 0 46000 4600 10
4 5 0 0 0 45000 4500 10
4 4 0 0 0 44000 4400 10
4 3 0 0 0 43000 4300 10
4 2 0 0 0 42000 4200 10
4 1 0 0 0 41000 4100 10
4 0 0 0 0 40000 4000 10
3 9 0 0 0 39000 3900 10
3 8 0 0 0 38000 3800 10
3 7 0 0 0 37000 3700 10
3 6 0 0 0 36000 3600 10
3 5 0 0 0 35000 3500 10
3 4 0 0 0 34000 3400 10
3 3 0 0 0 33000 3300 10
3 2 0 0 0 32000 3200 10
3 1 0 0 0 31000 3100 10
3 0 0 0 0 30000 3000 10
2 9 0 0 0 29000 2900 10
2 8 0 0 0 28000 2800 10
2 7 0 0 0 27000 2700 10
2 6 0 0 0 26000 2600 10
2 5 0 0 0 25000 2500 10
2 4 0 0 0 24000 2400 10
2 3 0 0 0 23000 2300 10
2 2 0 0 0 22000 2200 10
2 1 0 0 0 21000 2100 10
2 0 0 0 0 20000 2000 10
1 9 0 0 0 19000 1900 10
1 8 0 0 0 18000 1800 10
1 7 0 0 0 17000 1700 10
1 6 0 0 0 16000 1600 10
1 5 0 0 0 15000 1500 10
1 4 0 0 0 14000 1400 10
1 3 0 0 0 13000 1300 10
1 2 0 0 0 12000 1200 10
1 1 0 0 0 11000 1100 10
1 0 0 0 0 10000 1000 10

3. David @InfinitelyManic says:

9 9 0 0 0 99000 9900 10
9 8 0 0 0 98000 9800 10
9 7 0 0 0 97000 9700 10
9 6 0 0 0 96000 9600 10
9 5 0 0 0 95000 9500 10
9 4 0 0 0 94000 9400 10
9 3 0 0 0 93000 9300 10
9 2 0 0 0 92000 9200 10
9 1 0 0 0 91000 9100 10
9 0 0 0 0 90000 9000 10
8 9 0 0 0 89000 8900 10
8 8 0 0 0 88000 8800 10
8 7 0 0 0 87000 8700 10
8 6 0 0 0 86000 8600 10
8 5 0 0 0 85000 8500 10
8 4 0 0 0 84000 8400 10
8 3 0 0 0 83000 8300 10
8 2 0 0 0 82000 8200 10
8 1 0 0 0 81000 8100 10
8 0 0 0 0 80000 8000 10
7 9 0 0 0 79000 7900 10
7 8 0 0 0 78000 7800 10
7 7 0 0 0 77000 7700 10
7 6 0 0 0 76000 7600 10
7 5 0 0 0 75000 7500 10
7 4 0 0 0 74000 7400 10
7 3 0 0 0 73000 7300 10
7 2 0 0 0 72000 7200 10
7 1 0 0 0 71000 7100 10
7 0 0 0 0 70000 7000 10
6 9 0 0 0 69000 6900 10
6 8 0 0 0 68000 6800 10
6 7 0 0 0 67000 6700 10
6 6 0 0 0 66000 6600 10
6 5 0 0 0 65000 6500 10
6 4 0 0 0 64000 6400 10
6 3 0 0 0 63000 6300 10
6 2 0 0 0 62000 6200 10
6 1 0 0 0 61000 6100 10
6 0 0 0 0 60000 6000 10
5 9 0 0 0 59000 5900 10
5 8 0 0 0 58000 5800 10
5 7 0 0 0 57000 5700 10
5 6 0 0 0 56000 5600 10
5 5 0 0 0 55000 5500 10
5 4 0 0 0 54000 5400 10
5 3 0 0 0 53000 5300 10
5 2 0 0 0 52000 5200 10
5 1 0 0 0 51000 5100 10
5 0 0 0 0 50000 5000 10
4 9 0 0 0 49000 4900 10
4 8 0 0 0 48000 4800 10
4 7 0 0 0 47000 4700 10
4 6 0 0 0 46000 4600 10
4 5 0 0 0 45000 4500 10
4 4 0 0 0 44000 4400 10
4 3 0 0 0 43000 4300 10
4 2 0 0 0 42000 4200 10
4 1 0 0 0 41000 4100 10
4 0 0 0 0 40000 4000 10
3 9 0 0 0 39000 3900 10
3 8 0 0 0 38000 3800 10
3 7 0 0 0 37000 3700 10
3 6 0 0 0 36000 3600 10
3 5 0 0 0 35000 3500 10
3 4 0 0 0 34000 3400 10
3 3 0 0 0 33000 3300 10
3 2 0 0 0 32000 3200 10
3 1 0 0 0 31000 3100 10
3 0 0 0 0 30000 3000 10
2 9 0 0 0 29000 2900 10
2 8 0 0 0 28000 2800 10
2 7 0 0 0 27000 2700 10
2 6 0 0 0 26000 2600 10
2 5 0 0 0 25000 2500 10
2 4 0 0 0 24000 2400 10
2 3 0 0 0 23000 2300 10
2 2 0 0 0 22000 2200 10
2 1 0 0 0 21000 2100 10
2 0 0 0 0 20000 2000 10
1 9 0 0 0 19000 1900 10
1 8 0 0 0 18000 1800 10
1 7 0 0 0 17000 1700 10
1 6 0 0 0 16000 1600 10
1 5 0 0 0 15000 1500 10
1 4 0 0 0 14000 1400 10
1 3 0 0 0 13000 1300 10
1 2 0 0 0 12000 1200 10
1 1 0 0 0 11000 1100 10
1 0 0 0 0 10000 1000 10

4. David @InfinitelyManic says:

9 9 0 | 990 90 | 11
9 0 0 | 900 90 | 10
8 9 1 | 891 81 | 11
8 8 0 | 880 80 | 11
8 0 0 | 800 80 | 10
7 9 2 | 792 72 | 11
7 8 1 | 781 71 | 11
7 7 0 | 770 70 | 11
7 0 0 | 700 70 | 10
6 9 3 | 693 63 | 11
6 8 2 | 682 62 | 11
6 7 1 | 671 61 | 11
6 6 0 | 660 60 | 11
6 0 0 | 600 60 | 10
5 9 4 | 594 54 | 11
5 8 3 | 583 53 | 11
5 7 2 | 572 52 | 11
5 6 1 | 561 51 | 11
5 5 0 | 550 50 | 11
5 0 0 | 500 50 | 10
4 9 5 | 495 45 | 11
4 8 4 | 484 44 | 11
4 8 0 | 480 40 | 12
4 7 3 | 473 43 | 11
4 6 2 | 462 42 | 11
4 5 1 | 451 41 | 11
4 4 0 | 440 40 | 11
4 0 5 | 405 45 | 9
4 0 0 | 400 40 | 10
3 9 6 | 396 36 | 11
3 9 0 | 390 30 | 13
3 8 5 | 385 35 | 11
3 7 4 | 374 34 | 11
3 6 3 | 363 33 | 11
3 6 0 | 360 30 | 12
3 5 2 | 352 32 | 11
3 4 1 | 341 31 | 11
3 3 0 | 330 30 | 11
3 1 5 | 315 35 | 9
3 0 0 | 300 30 | 10
2 9 7 | 297 27 | 11
2 8 6 | 286 26 | 11
2 8 0 | 280 20 | 14
2 7 5 | 275 25 | 11
2 6 4 | 264 24 | 11
2 6 0 | 260 20 | 13
2 5 3 | 253 23 | 11
2 4 2 | 242 22 | 11
2 4 0 | 240 20 | 12
2 3 1 | 231 21 | 11
2 2 5 | 225 25 | 9
2 2 0 | 220 20 | 11
2 0 0 | 200 20 | 10
1 9 8 | 198 18 | 11
1 9 5 | 195 15 | 13
1 9 2 | 192 12 | 16
1 9 0 | 190 10 | 19
1 8 7 | 187 17 | 11
1 8 0 | 180 10 | 18
1 7 6 | 176 16 | 11
1 7 0 | 170 10 | 17
1 6 5 | 165 15 | 11
1 6 0 | 160 10 | 16
1 5 4 | 154 14 | 11
1 5 0 | 150 10 | 15
1 4 3 | 143 13 | 11
1 4 0 | 140 10 | 14
1 3 5 | 135 15 | 9
1 3 2 | 132 12 | 11
1 3 0 | 130 10 | 13
1 2 1 | 121 11 | 11
1 2 0 | 120 10 | 12
1 1 0 | 110 10 | 11
1 0 8 | 108 18 | 6
1 0 5 | 105 15 | 7
1 0 0 | 100 10 | 10