2021 AMC 10B Problems/Problem 13
Contents
Problem
Let be a positive integer and be a digit such that the value of the numeral in base equals , and the value of the numeral in base equals the value of the numeral in base six. What is
Solution 1
We can start by setting up an equation to convert base to base 10. To convert this to base 10, it would be Because it is equal to 263, we can set this equation to 263. Finally, subtract from both sides to get .
We can also set up equations to convert base and base 6 to base 10. The equation to covert base to base 10 is The equation to convert base 6 to base 10 is
Simplify so it becomes Setting the above equations equal to each other, we have Subtracting 4 from both sides gets
We can then use equations to solve for . Set equal to and solve to find that .
Plug back into the equation . Subtract 261 from both sides to get your final equation of We solve using the quadratic formula to find that the solutions are and Because the base must be positive,
Adding 2 to 9 gets
-Zeusthemoose (edited for readability) -solution corrected by Billowingsweater
Solution 2
is greater than when both are interpreted in base 10, so is less than . Some trial and error gives . in base 9 is , so the answer is .
-SmileKat32
Solution 3
We have Subtracting the 2nd from the 1st equation we get Thus we have Substituting into the first, we have Factoring, we have A digit cannot be negative, so we have Thus,
mathboy282 signing off ~ fixed by Lamboreghini
Solution 4
Find that using one of the methods above. Then we have that . We know that is an integer, so we can solve the equation (this is guaranteed to have a solution if we did this correctly). The prime factorization of is , so the corresponding factor pairs are , , and . If , then the equation is true, so .
Note: This solution provides an alternate way to complete the final part of the problem if, like me, you really dislike quadratics. ~ cxsmi
Video Solution (🚀 Under 4 min 🚀)
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Video Solution by OmegaLearn (Bases and System of Equations)
~ pi_is_3.14
Video Solution by TheBeautyofMath
https://youtu.be/L1iW94Ue3eI?t=880
~IceMatrix
Video Solution by Interstigation
~Interstigation
See Also
2021 AMC 10B (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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