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

 

Find other solutions.
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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