Serving the Quantitative Finance Community

 
User avatar
Chukchi
Topic Author
Posts: 0
Joined: December 15th, 2001, 3:43 am

12345678987654321

May 15th, 2011, 5:38 am

1. Prove that 12345678987654321 divides 10^111111111 - 1.2. Prove that 111111111^3 divides 10^12345678987654321 - 1.
 
User avatar
animeshsaxena
Posts: 18
Joined: June 19th, 2008, 2:56 pm

12345678987654321

May 15th, 2011, 10:12 am

12345678987654321 = 111111111 x 111111111This is from the pattern 121 = 11x11, 12321 = 111 x 1111So problem reduces to 10^(111111111)-1 divided by (11111111 x 111111111)10 power term -1 will give 9999.....repeated 11111111 times....*mistake here...for odd number it will be a problemAnd for any number 99 I can write 9 (10 + 1)9999 = 99 x (100+1) and so on...so change is...999=99*(10)+9*199999 = 999*(100)+99*19999999=9999*(1000)+999 (7 9's)the number 9 repeated 111111111 times is easily divisible by 111111111 x 111111111coz there are total 18 1's in divisor...and that many 9's can always be pulled out....
Last edited by animeshsaxena on June 18th, 2011, 10:00 pm, edited 1 time in total.
 
User avatar
QuantOrDie
Posts: 0
Joined: June 2nd, 2011, 2:23 am

12345678987654321

June 17th, 2011, 4:05 am

Last edited by QuantOrDie on June 16th, 2011, 10:00 pm, edited 1 time in total.