Consecutive integers; a(10^n * b + c) + 1 = (a + l)*b*c — Part 2

 
 
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 = 4

a \,(10^4 \; b \; + \; c) \; + \; 1 \; = \; (a + 1) \, b \, c        5000 < c < 9999

 
Here are the first few solutions:
 

26667 \; = \; 1 \; \times \; 26667
26668 \; = \; 2 \; \times \; 2 \; \times \; 6667

8345003 \; = \; 1 \; \times \; 8345003
8345004 \; = \; 2 \; \times \; 834 \; \times \; 5003

25015001 \; = \; 1 \; \times \; 25015001
25015002 \; = \; 2 \; \times \; 2501 \; \times \; 5001

753578 \; = \; 2 \; \times \; 376789
753579 \; = \; 3 \; \times \; 37 \; \times \; 6789

2473406 \; = \; 2 \; \times \; 1236703
2473407 \; = \; 3 \; \times \; 123 \; \times \; 6703

266713334 \; = \; 2 \; \times \; 133356667
266713335 \; = \; 3 \; \times \; 13335 \; \times \; 6667

168802503 \; = \; 3 \; \times \; 56267501
168802504 \; = \; 4 \; \times \; 5626 \; \times \; 7501

256072004 \; = \; 4 \; \times \; 64018001
256072005 \; = \; 5 \; \times \; 6401 \; \times \; 8001

10631607 \; = \; 7 \; \times \; 1518801
10631608 \; = \; 8 \; \times \; 151 \; \times \; 8801

178771271 \; = \; 7 \; \times \; 25538753
178771272 \; = \; 8 \; \times \; 2553 \; \times \; 8753

5689111112 \; = \; 8 \; \times \; 711138889
5689111113 \; = \; 9 \; \times \; 71113 \; \times \; 8889

729171009 \; = \; 9 \; \times \; 81019001
729171010 \; = \; 10 \; \times \; 8101 \; \times \; 9001

 
Find more solutions
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s