Difference between revisions of "2015 AIME I Problems/Problem 3"

(Solution 2 (Similar to 1))
Line 28: Line 28:
 
Observe that this is the same as <math>16p+1=n^3</math> for some integer <math>n</math>.
 
Observe that this is the same as <math>16p+1=n^3</math> for some integer <math>n</math>.
 
So:
 
So:
 +
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
16p &= n^3-1\\
 
16p &= n^3-1\\
Line 33: Line 34:
 
16p &= (n-1)(n^2+n+1)\\
 
16p &= (n-1)(n^2+n+1)\\
 
\end{align*}</cmath>
 
\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>).
 
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>.
 
Let <math>p=n-1</math>.
 
Then:
 
Then:
 +
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
p &= n-1\\
 
p &= n-1\\
Line 42: Line 45:
 
n(n+1) &= 15\\
 
n(n+1) &= 15\\
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
 
Which is obviously impossible for integer n. So <math>p=n^2+n+1</math> and
 
Which is obviously impossible for integer n. So <math>p=n^2+n+1</math> and
 +
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
16 &= n-1\\
 
16 &= n-1\\

Revision as of 12:50, 30 May 2020

Problem

There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

Solution 1

Let the positive integer mentioned be $a$, so that $a^3 = 16p+1$. Note that $a$ must be odd, because $16p+1$ is odd.

Rearrange this expression and factor the left side (this factoring can be done using $(a^3-b^3) = (a-b)(a^2+a b+b^2)$ or synthetic divison once it is realized that $a = 1$ is a root):

\begin{align*} a^3-1 &= 16p\\ (a-1)(a^2+a+1) &= 16p\\ \end{align*}

Because $a$ is odd, $a-1$ is even and $a^2+a+1$ is odd. If $a^2+a+1$ is odd, $a-1$ must be some multiple of $16$. However, for $a-1$ to be any multiple of $16$ other than $16$ would mean $p$ is not a prime. Therefore, $a-1 = 16$ and $a = 17$.

Then our other factor, $a^2+a+1$, is the prime $p$:

\begin{align*} (a-1)(a^2+a+1) &= 16p\\ (17-1)(17^2+17+1) &=16p\\ p = 289+17+1 &= \boxed{307} \end{align*}

Solution 2 (Similar to 1)

Observe that this is the same as $16p+1=n^3$ for some integer $n$. So:

\begin{align*} 16p &= n^3-1\\ 16p &= n^3-1^3\\ 16p &= (n-1)(n^2+n+1)\\ \end{align*}

Observe that either $p=n-1$ or $p=n^2+n+1$ because $p$ and $16$ share no factors ($p$ can't be $2$). Let $p=n-1$. Then:

\begin{align*} p &= n-1\\ 16 &= n^2+n+1\\ n^2+n &= 15\\ n(n+1) &= 15\\ \end{align*}

Which is obviously impossible for integer n. So $p=n^2+n+1$ and

\begin{align*} 16 &= n-1\\ n &= 17\\ p &= 17^2+17+1\\ p = 289+17+1 &= \boxed{307}\\ \end{align*} - firebolt360

Solution 3

Since $16p+1$ is odd, let $16p+1 = (2a+1)^3$. Therefore, $16p+1 = (2a+1)^3 = 8a^3+12a^2+6a+1$. From this, we get $8p=a(4a^2+6a+3)$. We know $p$ is a prime number and it is not an even number. Since $4a^2+6a+3$ is an odd number, we know that $a=8$.

Therefore, $p=4a^2+6a+3=4*8^2+6*8+3=\boxed{307}$.

Solution 4

Let $16p+1=a^3$. Realize that $a$ congruent to $1\mod 4$, so let $a=4n+1$. Expansion, then division by 4, gets $16n^3+12n^2+3n=4p$. Clearly $n=4m$ for some $m$. Substitution and another division by 4 gets $256m^3+48m^4+3m=p$. Since $p$ is prime and there is a factor of $m$ in the LHS, $m=1$. Therefore, $p=\boxed{307}$.

Solution 5

Notice that $16p+1$ must be in the form $(a+1)^3 = a^3 + 3a^2 + 3a + 1$. Thus $16p = a^3 + 3a^2 + 3a$, or $16p = a\cdot (a^2 + 3a + 3)$. Since $p$ must be prime, we either have $p = a$ or $a = 16$. Upon further inspection and/or using the quadratic formula, we can deduce $p \neq a$. Thus we have $a = 16$, and $p = 16^2 + 3\cdot 16 + 3 = \boxed{307}$.

Solution 6

Notice that the cube 16p+1 is equal to 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, $p=\boxed{307}$ is our only answer


pi_is_3.141

See also

2015 AIME I (ProblemsAnswer KeyResources)
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

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