## Consecutive integers; a(10^n * b + c) + 1 = (a + l)*b*c — Part 3

$x \; = \; a \,(10^{n} \; b \; + \; c)$,        $x + 1 \; = \; (a + 1) \, b \, c$

$a(10^n \; b \; + \; c) \; + \; 1 \; = \; (a + 1) \, b \, c$

where    $5 \; \times \; 10^{n-1} \; < \; c \; < \; 10^{n} \; - \; 1$

n = 5

$a \,(10^5 \; b \; + \; c) \; + \; 1 \; = \; (a + 1) \, b \, c$        $50000 < c < 99999$

Here are the first few solutions:

$266667 \; = \; 1 \; \times \; 266667$
$266668 \; = \; 2 \; \times \; 2 \; \times \; 66667$

$457143 \; = \; 1 \; \times \; 457143$
$457144 \; = \; 2 \; \times \; 4 \; \times \; 57143$

$1152381 \; = \; 1 \; \times \; 1152381$
$1152382 \; = \; 2 \; \times \; 11 \; \times \; 52381$

$119150021 \; = \; 1 \; \times \; 119150021$
$119150022 \; = \; 2 \; \times \; 1191 \; \times \; 50021$

$357250007 \; = \; 1 \; \times \; 357250007$
$357250008 \; = \; 2 \; \times \; 3572 \; \times \; 50007$

$833450003 \; = \; 1 \; \times \; 833450003$
$833450004 \; = \; 2 \; \times \; 8334 \; \times \; 50003$

$2500150001 \; = \; 1 \; \times \; 2500150001$
$2500150002 \; = \; 2 \; \times \; 25001 \; \times \; 50001$

$26667133334 \; = \; 2 \; \times \; 13333566667$
$26667133335 \; = \; 3 \; \times \; 133335 \; \times \; 66667$

$16875525003 \; = \; 3 \; \times \; 5625175001$
$16875525004 \; = \; 4 \; \times \; 56251 \; \times \; 75001$

$2327920044 \; = \; 4 \; \times \; 581980011$
$2327920045 \; = \; 5 \; \times \; 5819 \; \times \; 80011$

$25600720004 \; = \; 4 \; \times \; 6400180001$
$25600720005 \; = \; 5 \; \times \; 64001 \; \times \; 80001$

$879812927 \; = \; 7 \; \times \; 125687561$
$879812928 \; = \; 8 \; \times \; 1256 \; \times \; 87561$

$568891111112 \; = \; 8 \; \times \; 71111388889$
$568891111113 \; = \; 9 \; \times \; 711113 \; \times \; 88889$

$23340249 \; = \; 9 \; \times \; 2593361$
$23340250 \; = \; 10 \; \times \; 25 \; \times \; 93361$

$304112169 \; = \; 9 \; \times \; 33790241$
$304112170 \; = \; 10 \; \times \; 337 \; \times \; 90241$

$72901710009 \; = \; 9 \; \times \; 8100190001$
$72901710010 \; = \; 10 \; \times \; 81001 \; \times \; 90001$

$43291909110 \; = \; 10 \; \times \; 4329190911$
$43291909111 \; = \; 11 \; \times \; 43291 \; \times \; 90911$

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math grad - Interest: Number theory
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