Smarandache concatenated numbers - factors

Let SM1 = 1 SM2 = 12 SM3 = 123 SM4 = 1234 ... SM10 = 12345678910 SM11 = 1234567891011 etc.

a) Find n such that n is the least positive integer >= 1000

where 7 is a prime factor of SMn.

b) Find n such that n is the least positive integer where 7, 11, and 13 are all prime factors of SMn.

c) Find n such than n is the least positive integer >= 1000 where 101 is a prime factor of SMn.

(I solved with a QuickBasic program.)

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I hope these are correct, I did it rather quickly with sloppy code... =)

A) 1003 744 C) 10073

Anyone else get the same? Different?

1000 where 7 is a prime factor of SMn.<<

where 7, 11, and 13 are all prime factors of SMn.<<

1000 where 101 is a prime factor of SMn.<<

sloppy code... =)<

I get the same solutions you got for a) & b), Jeromy.

I get different for c).

I get 10073 + a number less than half of 101.

Bob

x x x x x x x x x x x x

Argh! Seems my big numbers package has a problem multiplying that huge number by a 6 digit number... See down below for my new answer to C...

10073 + 50 = 10123

I 'think' that's correct now. =)

Bob S. posted:

This can be done without a big numbers package.

The residue mod 7 (and separately mod 11, 13, and 101) of SM(n+1) can be calculated from the residue of SM(n). Doing so involves a small multiply and add, and a modulo operation. The best way I can think to describe this is to just show a little table (view with a monospace font).

+

Definitely more efficient than the brute force method. Easier to implement, too.

I did this one by hand, so I am probably way wrong. I don't think I've used pen and paper in over 10 years... I got: 38,888,888,899 digits. (Below table show my logic... Looks better with monospace font)

Low Bound - High Bound = Numbers * Dg = Total Digits

I get the following solutions for n, where n < 50000:

744 10134 11249 13390 13779 14527 14666 16140 17255 19396 19785 20533 20672 22146 23261 25402 25791 26539 26678 28152 29267 31408 31797 32545 32684 34158 35273 37414 37803 38551 38690 40164 41279 43420 43809 44557 44696 46170 47285 49426