Difference between revisions of "1990 AIME Problems/Problem 13"
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== Solution == | == Solution == | ||
− | When a number is multiplied by <math>9</math>, the number will gain a digit, except when the new number starts with a <math>9</math> | + | When a number is multiplied by <math>9</math>, the number will gain a digit, except when the new number starts with a <math>9</math>. 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 == |
Revision as of 19:35, 15 March 2017
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
When a number is multiplied by , the number will gain a digit, except when the new number starts with a . 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 |
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