Difference between revisions of "1990 AIME Problems/Problem 13"
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== Problem == | == Problem == | ||
| − | Let <math>T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}</math>. | + | Let <math>T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}</math>. Given that <math>9^{4000}_{}</math> has 3817 [[digit]]s and that its first (leftmost) digit is 9, how many [[element]]s of <math>T_{}^{}</math> have 9 as their leftmost digit? |
== Solution == | == Solution == | ||
| − | { | + | Whenever you multiply a number by <math>9</math>, the number will have an additional digit over the previous digit, with the exception when the new number starts with a <math>9</math>, when the number of digits remain the same. Since <math>9^{4000}</math> has 3816 digits more than <math>9^1</math>, exactly <math>4000 - (3817 - 1) = 184</math> numbers have 9 as their leftmost digits. |
== See also == | == See also == | ||
{{AIME box|year=1990|num-b=12|num-a=14}} | {{AIME box|year=1990|num-b=12|num-a=14}} | ||
Revision as of 13:16, 3 March 2007
Problem
Let 
. Given that 
has 3817 digits and that its first (leftmost) digit is 9, how many elements of 
have 9 as their leftmost digit?
Solution
Whenever you multiply a number by 
, the number will have an additional digit over the previous digit, with the exception when the new number starts with a , when the number of digits remain the same. Since 
has 3816 digits more than 
, exactly 
numbers have 9 as their leftmost digits.
See also
| 1990 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
