Difference between revisions of "2020 AIME II Problems/Problem 5"
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==Problem== | ==Problem== | ||
− | For each positive integer <math>n</math>, | + | For each positive integer <math>n</math>, let <math>f(n)</math> be the sum of the digits in the base-four representation of <math>n</math> and let <math>g(n)</math> be the sum of the digits in the base-eight representation of <math>f(n)</math>. For example, <math>f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}</math>, and <math>g(2020) = \text{the digit sum of }12_{\text{8}} = 3</math>. Let <math>N</math> be the least value of <math>n</math> such that the base-sixteen representation of <math>g(n)</math> cannot be expressed using only the digits <math>0</math> through <math>9</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. |
− | ==Solution== | + | |
− | Let's work backwards. | + | ==Solution 1== |
− | ==Video | + | Let's work backwards. The minimum base-sixteen representation of <math>g(n)</math> that cannot be expressed using only the digits <math>0</math> through <math>9</math> is <math>A_{16}</math>, which is equal to <math>10</math> in base 10. Thus, the sum of the digits of the base-eight representation of the sum of the digits of <math>f(n)</math> is <math>10</math>. The minimum value for which this is achieved is <math>37_8</math>. We have that <math>37_8 = 31</math>. Thus, the sum of the digits of the base-four representation of <math>n</math> is <math>31</math>. The minimum value for which this is achieved is <math>13,333,333,333_4</math>. We just need this value in base 10 modulo 1000. We get <math>13,333,333,333_4 = 3(1 + 4 + 4^2 + \dots + 4^8 + 4^9) + 4^{10} = 3\left(\dfrac{4^{10} - 1}{3}\right) + 4^{10} = 2*4^{10} - 1</math>. Taking this value modulo <math>1000</math>, we get the final answer of <math>\boxed{151}</math>. (If you are having trouble with this step, note that <math>2^{10} = 1024 \equiv 24 \pmod{1000}</math>) ~ TopNotchMath |
− | https://youtu.be/lTyiRQTtIZI | + | |
+ | ==Solution 2 (Official MAA)== | ||
+ | First note that if <math>h_b(s)</math> is the least positive integer whose digit sum, in some fixed base <math>b</math>, is <math>s</math>, then <math>h_b</math> is a strictly increasing function. This together with the fact that <math>g(N) \ge 10</math> shows that <math>f(N)</math> is the least positive integer whose base-eight digit sum is 10. Thus <math>f(N) = 37_\text{eight} = 31</math>, and <math>N</math> is the least positive integer whose base-four digit sum is <math>31.</math> Therefore <cmath> | ||
+ | N = 13333333333_\text{four} = 2\cdot4^{10} - 1 = 2\cdot1024^2 - 1</cmath> | ||
+ | <cmath>\equiv 2\cdot24^2 - 1 \equiv 151 \pmod{1000}.</cmath> | ||
+ | |||
+ | ==Video Solutions== | ||
+ | https://youtu.be/lTyiRQTtIZI | ||
− | |||
https://youtu.be/ZWe_99091e4 | https://youtu.be/ZWe_99091e4 | ||
− | ~ | + | == Video Solution by OmegaLearn == |
+ | https://youtu.be/ZhAZ1oPe5Ds?t=5032 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==See Also== | ==See Also== |
Latest revision as of 22:01, 23 March 2023
Contents
Problem
For each positive integer , let
be the sum of the digits in the base-four representation of
and let
be the sum of the digits in the base-eight representation of
. For example,
, and
. Let
be the least value of
such that the base-sixteen representation of
cannot be expressed using only the digits
through
. Find the remainder when
is divided by
.
Solution 1
Let's work backwards. The minimum base-sixteen representation of that cannot be expressed using only the digits
through
is
, which is equal to
in base 10. Thus, the sum of the digits of the base-eight representation of the sum of the digits of
is
. The minimum value for which this is achieved is
. We have that
. Thus, the sum of the digits of the base-four representation of
is
. The minimum value for which this is achieved is
. We just need this value in base 10 modulo 1000. We get
. Taking this value modulo
, we get the final answer of
. (If you are having trouble with this step, note that
) ~ TopNotchMath
Solution 2 (Official MAA)
First note that if is the least positive integer whose digit sum, in some fixed base
, is
, then
is a strictly increasing function. This together with the fact that
shows that
is the least positive integer whose base-eight digit sum is 10. Thus
, and
is the least positive integer whose base-four digit sum is
Therefore
Video Solutions
Video Solution by OmegaLearn
https://youtu.be/ZhAZ1oPe5Ds?t=5032
~ pi_is_3.14
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.