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

 
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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