2005 AMC 10A Problems/Problem 24

Problem

For each positive integer $n > 1$ , let $P(n)$ denote the greatest prime factor of $n$ . For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$ ?

$\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solution 1

If $P(n) = \sqrt{n}$ , then $n = p_{1}^{2}$ , where $p_{1}$ is a prime number.

If $P(n+48) = \sqrt{n+48}$ , then $n + 48$ is a square, but we know that n is $p_{1}^{2}$ .

This means we just have to check for squares of primes, add $48$ and look whether the root is a prime number. We can easily see that the difference between two consecutive square after $576$ is greater than or equal to $49$ , Hence we have to consider only the prime numbers till $23$ .

Squaring prime numbers below $23$ including $23$ we get the following list.

$4 , 9 , 25 , 49 , 121, 169 , 289 , 361 , 529$

But adding $48$ to a number ending with $9$ will result in a number ending with $7$ , but we know that a perfect square does not end in $7$ , so we can eliminate those cases to get the new list.

$4 , 25 , 121 , 361$

Adding $48$ , we get $121$ as the only possible solution. Hence the answer is $\boxed{\textbf{(B) }1}$ .

edited by mobius247

Note: Solution 1

Since all primes greater than $2$ are odd, we know that the difference between the squares of any two consecutive primes greater than $2$ is at least $(p+2)^2-p^2=4p+4$ , where p is the smaller of the consecutive primes. For $p>11$ , $4p+4>48$ . This means that the difference between the squares of any two consecutive primes both greater than $11$ is greater than $48$ , so $n$ and $n+48$ can't both be the squares of primes if $n=p^2$ and $p>11$ . So, we only need to check $n=2^2, 3^2, 5^2, 7^2,$ and $11^2$ .

~apsid

Video Solution

CHECK OUT Video Solution:https://youtu.be/IsqrsMkR-mA

~rudolf1279

Solution 2

If $P(n) = \sqrt{n}$ , then $n = p_{1}^{2}$ , where $p_{1}$ is a prime number.

If $P(n+48) = \sqrt{n+48}$ , then $n+48 = p_{2}^{2}$ , where $p_{2}$ is a different prime number.

So:

$p_{2}^{2} = n+48$

$p_{1}^{2} = n$

$p_{2}^{2} - p_{1}^{2} = 48$

$(p_{2}+p_{1})(p_{2}-p_{1})=48$

Since $p_{1} > 0$ : $(p_{2}+p_{1}) > (p_{2}-p_{1})$ .

Looking at pairs of divisors of $48$ , we have several possibilities to solve for $p_{1}$ and $p_{2}$ :

$(p_{2}+p_{1}) = 48$

$(p_{2}-p_{1}) = 1$

$p_{1} = \frac{47}{2}$

$p_{2} = \frac{49}{2}$

$(p_{2}+p_{1}) = 24$

$(p_{2}-p_{1}) = 2$

$p_{1} = 11$

$p_{2} = 13$

$(p_{2}+p_{1}) = 16$

$(p_{2}-p_{1}) = 3$

$p_{1} = \frac{13}{2}$

$p_{2} = \frac{19}{2}$

$(p_{2}+p_{1}) = 12$

$(p_{2}-p_{1}) = 4$

$p_{1} = 4$

$p_{2} = 8$

$(p_{2}+p_{1}) = 8$

$(p_{2}-p_{1}) = 6$

$p_{1} = 1$

$p_{2} = 7$

The only solution $(p_{1} , p_{2})$ where both numbers are primes is $(11,13)$ .

Therefore the number of positive integers $n$ that satisfy both statements is $\boxed{\textbf{(B) }1}.$

Video Solution 2

https://youtu.be/-8t5xaths_Q

~savannahsolver

2005 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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
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