Difference between revisions of "1996 AHSME Problems/Problem 14"
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| + | ==Problem== | ||
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| + | Let <math>E(n)</math> denote the sum of the even digits of <math>n</math>. For example, <math> E(5681) = 6+8 = 14 </math>. Find <math> E(1)+E(2)+E(3)+\cdots+E(100) </math> | ||
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| + | <math> \text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250 </math> | ||
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| + | ==Solution== | ||
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| + | The problem is asking for the sum of all the even digits in the numbers <math>1</math> to <math>100</math>. We can remove <math>100</math> from the list, add <math>00</math> to the list, and tack on some leading zeros to the single digit numbers without changing the sum of the even digits. This gives the list: | ||
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| + | <math>00, 01, 02, 03, ..., 10, 11, ..., 98, 99</math>. | ||
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| + | There are <math>2\cdot 100 = 200</math> digits on that list, and each digit appears <math>\frac{200}{10} = 20</math> times. | ||
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| + | Thus, each even digit appears <math>20</math> times, and the sum of all the even digits is <math>0 \cdot 20 + 2\cdot 20 + 4\cdot 20 + 6\cdot 20 + 8\cdot 20 = (0 + 2 + 4 + 6 + 8)\cdot 20 = 400</math>, and the correct answer is <math>\boxed{C}</math>. | ||
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==See also== | ==See also== | ||
{{AHSME box|year=1996|num-b=13|num-a=15}} | {{AHSME box|year=1996|num-b=13|num-a=15}} | ||
Revision as of 13:59, 19 August 2011
Problem
Let 
denote the sum of the even digits of 
. For example, 
. Find 
Solution
The problem is asking for the sum of all the even digits in the numbers 
to 
. We can remove from the list, add 
to the list, and tack on some leading zeros to the single digit numbers without changing the sum of the even digits. This gives the list:
.
There are 
digits on that list, and each digit appears 
times.
Thus, each even digit appears 
times, and the sum of all the even digits is 
, and the correct answer is 
.
See also
| 1996 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
