## Consecutive integers; a(10^n * b + c) + 1 = (a + l)*b*c — Part 4

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

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

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

n = 6

$a \,(10^6 \; b \; + \; c) \; + \; 1 \; = \; (a + 1) \, b \, c$        $500000 < c < 999999$

Here are the first few solutions:

$2666667 \; = \; 1 \; \times \; 2666667$
$2666668 \; = \; 2 \; \times \; 2 \; \times \; 666667$

$83334500003 \; = \; 1 \; \times \; 83334500003$
$83334500004 \; = \; 2 \; \times \; 83334 \; \times \; 500003$

$250001500001 \; = \; 1 \; \times \; 250001500001$
$250001500002 \; = \; 2 \; \times \; 250001 \; \times \; 500001$

$7714286 \; = \; 2 \; \times \; 3857143$
$7714287 \; = \; 3 \; \times \; 3 \; \times \; 857143$

$95352518 \; = \; 2 \; \times \; 47676259$
$95352519 \; = \; 3 \; \times \; 47 \; \times \; 676259$

$651336074 \; = \; 2 \; \times \; 325668037$
$651336075 \; = \; 3 \; \times \; 325 \; \times \; 668037$

$2743333982 \; = \; 2 \; \times \; 1371666991$
$2743333983 \; = \; 3 \; \times \; 1371 \; \times \; 666991$

$19187333426 \; = \; 2 \; \times \; 9593666713$
$19187333427 \; = \; 3 \; \times \; 9593 \; \times \; 666713$

$380955333338 \; = \; 2 \; \times \; 190477666669$
$380955333339 \; = \; 3 \; \times \; 190477 \; \times \; 666669$

$2666671333334 \; = \; 2 \; \times \; 1333335666667$
$2666671333335 \; = \; 3 \; \times \; 1333335 \; \times \; 666667$

$14769231 \; = \; 3 \; \times \; 4923077$
$14769232 \; = \; 4 \; \times \; 4 \; \times \; 923077$

$17647059 \; = \; 3 \; \times \; 5882353$
$17647060 \; = \; 4 \; \times \; 5 \; \times \; 882353$

$170280543 \; = \; 3 \; \times \; 56760181$
$170280544 \; = \; 4 \; \times \; 56 \; \times \; 760181$

$7640250663 \; = \; 3 \; \times \; 2546750221$
$7640250664 \; = \; 4 \; \times \; 2546 \; \times \; 750221$

$99269250051 \; = \; 3 \; \times \; 33089750017$
$99269250052 \; = \; 4 \; \times \; 33089 \; \times \; 750017$

$129812250039 \; = \; 3 \; \times \; 43270750013$
$129812250040 \; = \; 4 \; \times \; 43270 \; \times \; 750013$

$1687505250003 \; = \; 3 \; \times \; 562501750001$
$1687505250004 \; = \; 4 \; \times \; 562501 \; \times \; 750001$

$23809524 \; = \; 4 \; \times \; 5952381$
$23809525 \; = \; 5 \; \times \; 5 \; \times \; 952381$

$121911200084 \; = \; 4 \; \times \; 30477800021$
$121911200085 \; = \; 5 \; \times \; 30477 \; \times \; 800021$

$2560007200004 \; = \; 4 \; \times \; 640001800001$
$2560007200005 \; = \; 5 \; \times \; 640001 \; \times \; 800001$

$30857159142858 \; = \; 6 \; \times \; 5142859857143$
$30857159142859 \; = \; 7 \; \times \; 5142859 \; \times \; 857143$

$293258567 \; = \; 7 \; \times \; 41894081$
$293258568 \; = \; 8 \; \times \; 41 \; \times \; 894081$

$16708127247 \; = \; 7 \; \times \; 2386875321$
$16708127248 \; = \; 8 \; \times \; 2386 \; \times \; 875321$

$5359388125007 \; = \; 7 \; \times \; 765626875001$
$5359388125008 \; = \; 8 \; \times \; 765626 \; \times \; 875001$

$56888911111112 \; = \; 8 \; \times \; 7111113888889$
$56888911111113 \; = \; 9 \; \times \; 7111113 \; \times \; 888889$

$98901099 \; = \; 9 \; \times \; 10989011$
$98901100 \; = \; 10 \; \times \; 10 \; \times \; 989011$

$1970133579 \; = \; 9 \; \times \; 218903731$
$1970133580 \; = \; 10 \; \times \; 218 \; \times \; 903731$

$3374119539 \; = \; 9 \; \times \; 374902171$
$3374119540 \; = \; 10 \; \times \; 374 \; \times \; 902171$

$80126100819 \; = \; 9 \; \times \; 8902900091$
$80126100820 \; = \; 10 \; \times \; 8902 \; \times \; 900091$

$177821100369 \; = \; 9 \; \times \; 19757900041$
$177821100370 \; = \; 10 \; \times \; 19757 \; \times \; 900041$

$7290017100009 \; = \; 9 \; \times \; 810001900001$
$7290017100010 \; = \; 10 \; \times \; 810001 \; \times \; 900001$

$119909910 \; = \; 10 \; \times \; 11990991$
$119909911 \; = \; 11 \; \times \; 11 \; \times \; 990991$

$819019091010 \; = \; 10 \; \times \; 81901909101$
$819019091011 \; = \; 11 \; \times \; 81901 \; \times \; 909101$

$90909119090910 \; = \; 10 \; \times \; 9090911909091$
$90909119090911 \; = \; 11 \; \times \; 9090911 \; \times \; 909091$

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