Difference between revisions of "2015 AIME I Problems/Problem 3"
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There is a prime number <math>p</math> such that <math>16p+1</math> is the cube of a positive integer. Find <math>p</math>. | There is a prime number <math>p</math> such that <math>16p+1</math> is the cube of a positive integer. Find <math>p</math>. | ||
+ | ==Video Solution For Problems 1-3== | ||
+ | https://www.youtube.com/watch?v=5HAk-6qlOH0 | ||
− | ==Solution== | + | == Video Solution by OmegaLearn == |
+ | https://youtu.be/3bRjcrkd5mQ?t=1096 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution by WhyMath== | ||
+ | https://youtu.be/vajttONefxs | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Solution 1== | ||
Let the positive integer mentioned be <math>a</math>, so that <math>a^3 = 16p+1</math>. Note that <math>a</math> must be odd, because <math>16p+1</math> is odd. | Let the positive integer mentioned be <math>a</math>, so that <math>a^3 = 16p+1</math>. Note that <math>a</math> must be odd, because <math>16p+1</math> is odd. | ||
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\end{align*}</cmath> | \end{align*}</cmath> | ||
− | == | + | ==Solution 2 (Similar to 1)== |
+ | |||
+ | Observe that this is the same as <math>16p+1=n^3</math> for some integer <math>n</math>. | ||
+ | So: | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | 16p &= n^3-1\\ | ||
+ | 16p &= n^3-1^3\\ | ||
+ | 16p &= (n-1)(n^2+n+1)\\ | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | Observe that either <math>p=n-1</math> or <math>p=n^2+n+1</math> because <math>p</math> and <math>16</math> share no factors (<math>p</math> can't be <math>2</math>). | ||
+ | Let <math>p=n-1</math>. | ||
+ | Then: | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | p &= n-1\\ | ||
+ | 16 &= n^2+n+1\\ | ||
+ | n^2+n &= 15\\ | ||
+ | n(n+1) &= 15\\ | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | Which is impossible for integer n. So <math>p=n^2+n+1</math> and | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | 16 &= n-1\\ | ||
+ | n &= 17\\ | ||
+ | p &= 17^2+17+1\\ | ||
+ | p = 289+17+1 &= \boxed{307}\\ | ||
+ | \end{align*}</cmath> - firebolt360 | ||
+ | |||
+ | ==Solution 3== | ||
Since <math>16p+1</math> is odd, let <math>16p+1 = (2a+1)^3</math>. Therefore, <math>16p+1 = (2a+1)^3 = 8a^3+12a^2+6a+1</math>. From this, we get <math>8p=a(4a^2+6a+3)</math>. We know <math>p</math> is a prime number and it is not an even number. Since <math>4a^2+6a+3</math> is an odd number, we know that <math>a=8</math>. | Since <math>16p+1</math> is odd, let <math>16p+1 = (2a+1)^3</math>. Therefore, <math>16p+1 = (2a+1)^3 = 8a^3+12a^2+6a+1</math>. From this, we get <math>8p=a(4a^2+6a+3)</math>. We know <math>p</math> is a prime number and it is not an even number. Since <math>4a^2+6a+3</math> is an odd number, we know that <math>a=8</math>. | ||
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Therefore, <math>p=4a^2+6a+3=4*8^2+6*8+3=\boxed{307}</math>. | Therefore, <math>p=4a^2+6a+3=4*8^2+6*8+3=\boxed{307}</math>. | ||
− | == | + | ==Solution 4== |
− | Let <math>16p+1=a^3</math>. Realize that <math>a</math> | + | |
+ | Let <math>16p+1=a^3</math>. Realize that <math>a</math> congruent to <math>1\mod 4</math>, so let <math>a=4n+1</math>. Expansion, then division by 4, gets <math>16n^3+12n^2+3n=4p</math>. Clearly <math>n=4m</math> for some <math>m</math>. Substitution and another division by 4 gets <math>256m^3+48m^4+3m=p</math>. Since <math>p</math> is prime and there is a factor of <math>m</math> in the LHS, <math>m=1</math>. Therefore, <math>p=\boxed{307}</math>. | ||
+ | |||
+ | ==Solution 5== | ||
+ | |||
+ | Notice that <math>16p+1</math> must be in the form <math>(a+1)^3 = a^3 + 3a^2 + 3a + 1</math>. Thus <math>16p = a^3 + 3a^2 + 3a</math>, or <math>16p = a\cdot (a^2 + 3a + 3)</math>. Since <math>p</math> must be prime, we either have <math>p = a</math> or <math>a = 16</math>. Upon further inspection and/or using the quadratic formula, we can deduce <math>p \neq a</math>. Thus we have <math>a = 16</math>, and <math>p = 16^2 + 3\cdot 16 + 3 = \boxed{307}</math>. | ||
+ | |||
+ | ==Solution 6== | ||
+ | Notice that the cube 16p+1 is congruent to 1 mod 16. The only cubic numbers that leave a residue of 1 mod 16 are 1 and 15. | ||
+ | Case one: The cube is of the form 16k+1-->Plugging this in, and taking note that p is prime and has only 1 factor gives p=307 | ||
+ | Case two: The cube is of the form 16k+15--> Plugging this in, we quickly realize that this case is invalid, as that implies p is even, and p=2 doesn't work here | ||
+ | |||
+ | Hence, <math>p=\boxed{307}</math> is our only answer | ||
+ | |||
+ | pi_is_3.141 | ||
+ | |||
+ | ==Solution 7== | ||
+ | If <math>16p+1 =k</math>, we have <math>k \equiv 1 \mod 16</math>, so <math>k^3 \equiv 1 \mod 16</math>. If <math>k=1</math> we have <math>p=0</math>, which is not prime. If <math>k=17</math> we have <math>16p+1=4913</math>, or <math>p=\boxed{307}</math> | ||
+ | |||
+ | ==Solution 8 (Pattern Recognition)== | ||
+ | Notice that: | ||
+ | <cmath>\begin{align*} | ||
+ | 1^3 &= 0 +1\\ | ||
+ | 2^3 &= 1*7+1 \\ | ||
+ | 3^3 &= 2*13+1\\ | ||
+ | 4^3 &= 3*21+1\\ | ||
+ | 5^3 &= 4*31+1\\ | ||
+ | 6^3 &= 5*43+1 | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | Here, we can see a clear pattern that <math>n^3=(n-1)p+1</math>, where <math>p</math> is some positive (not necessarily prime) integer. Hence, the equation <math>16p+1=a^3</math> can interpret as <math>17^3 = 16p+1</math>. Solving it, we got <math>p=307</math>. After checking all possible divisors, we will find that <math>307</math> is prime. Hence, we got <math>p=\boxed{307}</math>. | ||
+ | |||
+ | ==Solution 9 (Slightly Different Modular Arithmetic)== | ||
+ | |||
+ | We see that <math>16p+1=n^3</math> for a positive integer <math>n</math>. Subtracting <math>1</math>, we can turn this equation into a modular congruence, since <math>n^3-1</math> must be a multiple of <math>16</math>. | ||
+ | |||
+ | Since <math>n^3-1\equiv0\pmod{16}</math>, <math>n^3\equiv1\pmod{16}</math>. We observe that <math>n=1</math> is a solution to this congruence, which doesn't work. The next, or most obvious number to try is <math>n=17</math>. Plugging this in to our original equation, we get | ||
+ | |||
+ | <math>17^3=16p+1</math>, yielding <math>p=\boxed{307}</math>, which is prime. | ||
+ | |||
+ | |||
+ | -among us (countmath1) | ||
+ | |||
+ | ==Solution 10 (Lengthy Brute Force)== | ||
+ | Recognizing that AIME answers are <math>0</math> through <math>999</math>, the numbers whose cube could even be in contention to be equal to <math>16p + 1</math> are <math>4-25</math>. The cubes of <math>1-3</math> are all below <math>17</math>. We might consider <math>1</math>, but that would result in a <math>p</math> of <math>0</math>, which is not prime and does not follow our given conditions. We can also see that <math>16p + 1</math> must be an odd number. Hence, we brute force by looking at all cubes of odd numbers in <math>5-25</math> until we get that the cube of <math>17</math>, <math>4913</math>, works when <math>p=\boxed{307}</math> | ||
+ | |||
+ | Solution by: armang32324 | ||
== See also == | == See also == |
Latest revision as of 23:19, 23 January 2024
Contents
- 1 Problem
- 2 Video Solution For Problems 1-3
- 3 Video Solution by OmegaLearn
- 4 Video Solution by WhyMath
- 5 Solution 1
- 6 Solution 2 (Similar to 1)
- 7 Solution 3
- 8 Solution 4
- 9 Solution 5
- 10 Solution 6
- 11 Solution 7
- 12 Solution 8 (Pattern Recognition)
- 13 Solution 9 (Slightly Different Modular Arithmetic)
- 14 Solution 10 (Lengthy Brute Force)
- 15 See also
Problem
There is a prime number such that
is the cube of a positive integer. Find
.
Video Solution For Problems 1-3
https://www.youtube.com/watch?v=5HAk-6qlOH0
Video Solution by OmegaLearn
https://youtu.be/3bRjcrkd5mQ?t=1096
~ pi_is_3.14
Video Solution by WhyMath
~savannahsolver
Solution 1
Let the positive integer mentioned be , so that
. Note that
must be odd, because
is odd.
Rearrange this expression and factor the left side (this factoring can be done using or synthetic divison once it is realized that
is a root):
Because is odd,
is even and
is odd. If
is odd,
must be some multiple of
. However, for
to be any multiple of
other than
would mean
is not a prime. Therefore,
and
.
Then our other factor, , is the prime
:
Solution 2 (Similar to 1)
Observe that this is the same as for some integer
.
So:
Observe that either or
because
and
share no factors (
can't be
).
Let
.
Then:
Which is impossible for integer n. So and
- firebolt360
Solution 3
Since is odd, let
. Therefore,
. From this, we get
. We know
is a prime number and it is not an even number. Since
is an odd number, we know that
.
Therefore, .
Solution 4
Let . Realize that
congruent to
, so let
. Expansion, then division by 4, gets
. Clearly
for some
. Substitution and another division by 4 gets
. Since
is prime and there is a factor of
in the LHS,
. Therefore,
.
Solution 5
Notice that must be in the form
. Thus
, or
. Since
must be prime, we either have
or
. Upon further inspection and/or using the quadratic formula, we can deduce
. Thus we have
, and
.
Solution 6
Notice that the cube 16p+1 is congruent to 1 mod 16. The only cubic numbers that leave a residue of 1 mod 16 are 1 and 15. Case one: The cube is of the form 16k+1-->Plugging this in, and taking note that p is prime and has only 1 factor gives p=307 Case two: The cube is of the form 16k+15--> Plugging this in, we quickly realize that this case is invalid, as that implies p is even, and p=2 doesn't work here
Hence, is our only answer
pi_is_3.141
Solution 7
If , we have
, so
. If
we have
, which is not prime. If
we have
, or
Solution 8 (Pattern Recognition)
Notice that:
Here, we can see a clear pattern that , where
is some positive (not necessarily prime) integer. Hence, the equation
can interpret as
. Solving it, we got
. After checking all possible divisors, we will find that
is prime. Hence, we got
.
Solution 9 (Slightly Different Modular Arithmetic)
We see that for a positive integer
. Subtracting
, we can turn this equation into a modular congruence, since
must be a multiple of
.
Since ,
. We observe that
is a solution to this congruence, which doesn't work. The next, or most obvious number to try is
. Plugging this in to our original equation, we get
, yielding
, which is prime.
-among us (countmath1)
Solution 10 (Lengthy Brute Force)
Recognizing that AIME answers are through
, the numbers whose cube could even be in contention to be equal to
are
. The cubes of
are all below
. We might consider
, but that would result in a
of
, which is not prime and does not follow our given conditions. We can also see that
must be an odd number. Hence, we brute force by looking at all cubes of odd numbers in
until we get that the cube of
,
, works when
Solution by: armang32324
See also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.