Difference between revisions of "2009 AMC 10A Problems/Problem 5"
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==Problem== | ==Problem== | ||
| − | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>What is the sum of the digits of the square of <math>111,111,111</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>What is the sum of the digits of the square of <math>\text 111,111,111</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
<math>\mathrm{(A)}\ 18\qquad\mathrm{(B)}\ 27\qquad\mathrm{(C)}\ 45\qquad\mathrm{(D)}\ 63\qquad\mathrm{(E)}\ 81</math> | <math>\mathrm{(A)}\ 18\qquad\mathrm{(B)}\ 27\qquad\mathrm{(C)}\ 45\qquad\mathrm{(D)}\ 63\qquad\mathrm{(E)}\ 81</math> | ||
Revision as of 18:32, 5 February 2018
Problem
What is the sum of the digits of the square of 
?
Solution 1
Using the standard multiplication algorithm, 
whose digit sum is 
(I hope you didn't seriously multiply it out right... ;) )
Solution 2
Note that:
We see a pattern and find that 
whose digit sum is
Solution 3
You can see that
can be written as
, which is 
.
We can apply the same fact into 111,111,111, receiving
whose digits sum up to
See also
| 2009 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
