2005 AMC 10A Problems/Problem 24
Problem
For each positive integer , let denote the greatest prime factor of . For how many positive integers is it true that both and ?
Solution 1
If , then , where is a prime number.
If , then is a square, but we know that n is .
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.
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.
Adding 48, we get 121 as the only possible solution.
Hence the answer is (B).
The only positive integer that satisfies both requirements is 11.
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 , where p is the smaller of the consecutive primes. For , . This means that the difference between the squares of any two consecutive primes both greater than 11 is greater than 48, so and can't both be the squares of primes if and p>11. So, we only need to check .
~apsid
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~rudolf1279
Solution 2
If , then , where is a prime number.
If , then , where is a different prime number.
So:
Since : .
Looking at pairs of divisors of , we have several possibilities to solve for and :
The only solution where both numbers are primes is .
Therefore the number of positive integers that satisfy both statements is
See Also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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