A number theory problem by Clyde Sale

Number Theory Level 3

If 10^{2012} - 2012 is written out in full, the sum of its digits is _ .

(This is an AMC 2012 question.)


The answer is 18104.

Clyde Sale by Clyde Sale , 27, Philippines

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

4 solutions

Mohinder Goyal
Jul 17, 2014

10^{2012} will be a 2013 digit number with first digit as 1 followed by all zeros. Last 4 digits after subtraction will be (7988) there will be 2012 digits in the answer after subtraction. first 2008 digits of the answer will be 9

so sum of digits = 9*(2008) + 7 + 9 + 8 + 8 = 18104

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Bill Bell
Jul 25, 2014

I learned to program because I was never able to get the same result twice for any arithmetic or algebra expression. Brute force in Python:

>>> sum([int(c) for c in str(10**2012-2012)])

18104

However, I would just like to say that I tried subtracting 2012 from 1000000 first!

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Lachlan Kuhr
Jan 15, 2015

10^2012 will have be the number starting with 1 followed by 2012 zeros. Since you are subtracting 2012 off this number the result will be 999...999 7988 . Those last four digits will are the only variance in the repition of the digit 9.

Thus, sum of digits = 9*(2008) + 7 + 9 + 8 +8 = 18104.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ivan Martinez
Sep 28, 2014

-10^{2012}-2012 will have 2012 digits. -10^{2012}-2012 = 9999.....99997988 -As the number will have 2012 digits, the number of nines is:2012-4 = 2008 -(2008)*9 + 7 + 9 + 8 + 8 = 1804

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...