2003 AMC 10A Problems/Problem 20
Contents
Problem 20
A base-10 three digit number is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of are both three-digit numerals?
Solution
If we explore a similar problem: Which positive intergers have 3 digits in base 10? The smallest one ranges from 100-999, or 10^2 --> 10^3-1 Therefore, The smallest base-11 number that has 3 digits in base-10 is which is . because 11^2
The largest number in base-9 that has 3 digits in base-10 is Alternatively, you can do
Hence, all numbers that will have 3 digits in base-9, 10, and 11 will be between and , thus the total amount of numbers that will have 3 digits in base-9, 10, and 11 is
There are 900 possible 3 digit numbers in base 10, because it is 9 possibilities for the hundreds digit, 10 for the tens digit, and 10 fo the units digits, so its 9x10x10 which is .
Hence, the answer is
~CharmaineMa07292010
Video Solution by OmegaLearn
https://youtu.be/SCGzEOOICr4?t=596
~ pi_is_3.14
Video Solution
~IceMatrix
Video Solution by WhyMath
~savannahsolver
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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All AMC 10 Problems and Solutions |
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