2009 AMC 10A Problems/Problem 5
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 -- Nonrigorous solution
We note that
,
,
,
and .
We can clearly see the pattern: If is , with ones (and for the sake of simplicity, assume that ), then the sum of the digits of is
Aha! We know that has digits, so its digit sum is .
Solution 3
We see that can be written as .
We can apply this strategy to find , as seen below.
The digit sum is thus .
See also
2009 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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