2. What is the greatest possible length of a positive integer less than 1,000?
Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)
A. 10
B. 9
C. 8
D. 7
E. 6
Question 2
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- GMATGuruNY
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To MAXIMIZE the length of n, the prime-factorization of n must include AS MANY PRIME FACTORS AS POSSIBLE.oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?
Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)
A. 10
B. 9
C. 8
D. 7
E. 6
For this reason, we must MINIMIZE the value of each prime factor.
Since 2 is the smallest prime factor, the prime-factorization of n must include as many 2's as possible.
If n = 2*2*2*2*2*2*2*2*2 = 512, then the prime-factorization of n has nine prime factors, yielding a length of 9.
If the prime-factorization of n includes any more 2's, the value of n will exceed 1000.
Since the prime-factorization n cannot include more 2's than the 9 values in red, the greatest possible length of n = 9.
The correct answer is B.
Last edited by GMATGuruNY on Fri Sep 18, 2015 12:50 pm, edited 2 times in total.
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GMATGuruNY wrote:To MAXIMIZE the length, we must MINIMIZE the value of each prime factor.oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?
Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)
A. 10
B. 9
C. 8
D. 7
E. 6
Thus, the prime-factorization of n must includes as many 2's as possible:
2*2*2*2*2*2*2*2*2 = 512.
If any other prime factors are included, the value n will exceed 1000.
Since the prime-factoization above is composed of nine 2's, the greatest possible length of n = 9.
The correct answer is B.
Why use 2? I dont understand why.
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Hi oquiella,
Since the question asks for the GREATEST possible "length", we need to get as many primes into the product as possible (AND keep the total less than 1,000). Consider the following:
If we used ONLY 5s....
(5)(5)(5)(5) = 625.....but this number only has a "length" of 4 (since there are only 4 primes)
If we used ONLY 3s...
(3)(3)(3)(3)(3)(3) = 729...but this number only has a "length" of 6 (since there are only 6 primes)
By using ONLY 2s, we end up with a "length" of 9 (and THAT is the greatest length possible; no matter how many other examples you might try, you won't end up with a "length" greater than 9).
GMAT assassins aren't born, they're made,
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Since the question asks for the GREATEST possible "length", we need to get as many primes into the product as possible (AND keep the total less than 1,000). Consider the following:
If we used ONLY 5s....
(5)(5)(5)(5) = 625.....but this number only has a "length" of 4 (since there are only 4 primes)
If we used ONLY 3s...
(3)(3)(3)(3)(3)(3) = 729...but this number only has a "length" of 6 (since there are only 6 primes)
By using ONLY 2s, we end up with a "length" of 9 (and THAT is the greatest length possible; no matter how many other examples you might try, you won't end up with a "length" greater than 9).
GMAT assassins aren't born, they're made,
Rich
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In order to maximize the "length," we need to minimize the values of the prime factors of n. Since 2 is the smallest prime, let's see how many factors of 2 we can use to get a product less than 1000. Since 2^9 = 512, we see that the largest possible length of a positive integer less than 1000 is 9.oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?
Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)
A. 10
B. 9
C. 8
D. 7
E. 6
Answer: B
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