Difference between revisions of "2021 AMC 10B Problems/Problem 13"
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<math>\textbf{(A)} ~10 \qquad\textbf{(B)} ~11 \qquad\textbf{(C)} ~13 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16</math> | <math>\textbf{(A)} ~10 \qquad\textbf{(B)} ~11 \qquad\textbf{(C)} ~13 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16</math> | ||
− | ==Solution== | + | ==Solution 1== |
We can start by setting up an equation to convert <math>\underline{32d}</math> base <math>n</math> to base 10. To convert this to base 10, it would be <math>3{n}^2+2n+d.</math> Because it is equal to 263, we can set this equation to 263. Finally, subtract <math>d</math> from both sides to get <math>3{n}^2+2n = 263-d</math>. | We can start by setting up an equation to convert <math>\underline{32d}</math> base <math>n</math> to base 10. To convert this to base 10, it would be <math>3{n}^2+2n+d.</math> Because it is equal to 263, we can set this equation to 263. Finally, subtract <math>d</math> from both sides to get <math>3{n}^2+2n = 263-d</math>. | ||
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We can also set up equations to convert <math>\underline{324}</math> base <math>n</math> and <math>\underline{11d1}</math> base 6 to base 10. The equation to covert <math>\underline{324}</math> base <math>n</math> to base 10 is <math>3{n}^2+2n+4.</math> The equation to convert <math>\underline{11d1}</math> base 6 to base 10 is <math>{6}^3+{6}^2+6d+1.</math> | We can also set up equations to convert <math>\underline{324}</math> base <math>n</math> and <math>\underline{11d1}</math> base 6 to base 10. The equation to covert <math>\underline{324}</math> base <math>n</math> to base 10 is <math>3{n}^2+2n+4.</math> The equation to convert <math>\underline{11d1}</math> base 6 to base 10 is <math>{6}^3+{6}^2+6d+1.</math> | ||
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Simplify <math>{6}^3+{6}^2+6d+1</math> so it becomes <math>6d+253.</math> Setting the above equations equal to each other, we have | Simplify <math>{6}^3+{6}^2+6d+1</math> so it becomes <math>6d+253.</math> Setting the above equations equal to each other, we have | ||
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<cmath>3{n}^2+2n = 6d+249</cmath> | <cmath>3{n}^2+2n = 6d+249</cmath> | ||
to solve for <math>d</math>. Set <math>263-d</math> equal to <math>6d+249</math> and solve to find that <math>d=2</math>. | to solve for <math>d</math>. Set <math>263-d</math> equal to <math>6d+249</math> and solve to find that <math>d=2</math>. | ||
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Plug <math>d=2</math> back into the equation <math>3{n}^2+2n = 263-d</math>. Subtract 261 from both sides to get your final equation of <math>3{n}^2+2n-261 = 0.</math> Solve using the quadratic formula to find that the solutions are 9 and -10. Because the base must be positive, <math>n=9.</math> | Plug <math>d=2</math> back into the equation <math>3{n}^2+2n = 263-d</math>. Subtract 261 from both sides to get your final equation of <math>3{n}^2+2n-261 = 0.</math> Solve using the quadratic formula to find that the solutions are 9 and -10. Because the base must be positive, <math>n=9.</math> | ||
− | Adding 2 to 9 gets <math>\boxed{\textbf{(B)}11}</math | + | Adding 2 to 9 gets <math>\boxed{\textbf{(B)} ~11}</math> |
− | |||
-Zeusthemoose (edited for readability) | -Zeusthemoose (edited for readability) | ||
==Solution 2== | ==Solution 2== | ||
− | <math>32d</math> is greater than <math>263</math> when both are interpreted in base 10, so <math>n</math> is less than <math>10</math>. Some trial and error gives <math>n=9</math>. <math>263</math> in base 9 is <math>322</math>, so the answer is <math>9+2=\boxed{\textbf{(B)}11}</math>. | + | <math>32d</math> is greater than <math>263</math> when both are interpreted in base 10, so <math>n</math> is less than <math>10</math>. Some trial and error gives <math>n=9</math>. <math>263</math> in base 9 is <math>322</math>, so the answer is <math>9+2=\boxed{\textbf{(B)} ~11}</math>. |
-SmileKat32 | -SmileKat32 |
Revision as of 06:50, 12 June 2021
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
Solve using the quadratic formula to find that the solutions are 9 and -10. Because the base must be positive,
Adding 2 to 9 gets
-Zeusthemoose (edited for readability)
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.