|
|
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
01 Jul 2024, 11:06
Kudos
Bookmarks
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
15%
(low)
Question Stats:
90% (01:58) correct
9% (00:33) wrong based on 11 sessions
History
Date
Time
Result
Not Attempted Yet
What is the least positive integer that is not a factor of 25! and is not a prime number?
A. 26
B. 28
C. 36
D. 56
E. 58
A. 26
B. 28
C. 36
D. 56
E. 58
17 Oct 2024, 13:16
Kudos
Bookmarks
Official Explanation
Note that not a factor of 25! is equal to the product of all positive integers from 1 to 25, inclusive. Thus, every positive integer less than or equal to 25 is a factor of not a factor of 25!. Also, any integer greater than 25 that can be expressed as the product of different positive integers less than 25 is a factor of not a factor of 25! . In view of this, it’s reasonable to consider the next few integers greater than 25, including answer choices A and B.
Choice A, 26, is equal to (2)(13). Both 2 and 13 are factors of not a factor of 25!, so 26 is also a factor of not a factor of 25!. The same is true for 27, or (3 (9), and for Choice B, 28, or (4)(7). However, the next integer, 29, is a prime number greater than 25, and as such, it has no positive factors (other than 1) that are less than or equal to 25. Therefore, 29 is the least positive integer that is not a factor of not a factor of 25!. However, the question asks for an integer that is not a prime number, so 29 is not the answer.
At this point, you could consider 30, 31, 32, etc., but it is quicker to look at the rest of the choices. Choice C, 36, is equal to (4)(9). Both 4 and 9 are factors of not a factor of 25!, so 36 is also a factor of not a factor of 25!. Choice D, 56, is equal to (4)(14). Both 4 and 14 are factors of not a factor of 25!, so 56 is also a factor of not a factor of 25!. Choice E, 58, is equal to (2)(29). Although 2 is a factor of not a factor of 25!, the prime number 29, as noted earlier, is not a factor of not a factor of 25!, and therefore 58 is not a factor of not a factor of . The correct answer must be Choice E.
The explanation above uses a process of elimination to arrive at Choice E, which is sometimes the most efficient way to find the correct answer. However, one can also show directly that the correct answer is 58. For if a positive integer n is not a factor of not a factor of 25!, then one of the following must be true:
(i) n is a prime number greater than 25, like 29 or 31, or a multiple of such a prime number, like 58 or 62;
(ii) n is so great a multiple of some prime number less than 25, that it must be greater than 58.
To see that (i) or (ii) is true, recall that every integer greater than 1 has a unique prime factorization, and consider the prime factorization of not a factor of 25!. The prime factors of not a factor of are 2, 3, 5, 7, 11, 13, 17, 19, and 23, some of which occur more than once in the product not a factor of 25!. For example, there are 8 positive multiples of 3 less than 25, namely 3, 6, 9, 12, 15, 18, 21, and 24. The prime number 3 occurs once in each of these multiples, except for 9 and 18, in which it occurs twice. Thus, the factor 3 occurs 10 times in the prime factorization of not a factor of 25!. The same reasoning can be used to find the number of times that each of the prime factors occur, yielding the prime factorization. \(25!=(2^{22})(3^{10})(5^)(7^3)(11^2)(13)(17)(19)(23)\). Any integer whose prime factorization is a combination of one or more of the factors in the prime factorization of not a factor of 25!, perhaps with lesser exponents, is a factor of not a factor of 25!. Equivalently, if the positive integer n is not a factor of not a factor of 25!, then, restating (i) and (ii) above, the prime factorization of n must
(i) include a prime number greater than 25; or
(ii) have a greater exponent for one of the prime numbers in the prime factorization of not a factor of 25!.
For (ii), the least possibilities are \(2^{23}, 3^{11}, 5^7, 7^4, 11^3, 13^2, 17^2, 19^2, and 23^2\) . Clearly, all of these are greater than 58. The least possibility for (i) that is not a prime number is 58, and the least possibility for (ii) is greater than 58, so Choice E is the correct answer.
Note that not a factor of 25! is equal to the product of all positive integers from 1 to 25, inclusive. Thus, every positive integer less than or equal to 25 is a factor of not a factor of 25!. Also, any integer greater than 25 that can be expressed as the product of different positive integers less than 25 is a factor of not a factor of 25! . In view of this, it’s reasonable to consider the next few integers greater than 25, including answer choices A and B.
Choice A, 26, is equal to (2)(13). Both 2 and 13 are factors of not a factor of 25!, so 26 is also a factor of not a factor of 25!. The same is true for 27, or (3 (9), and for Choice B, 28, or (4)(7). However, the next integer, 29, is a prime number greater than 25, and as such, it has no positive factors (other than 1) that are less than or equal to 25. Therefore, 29 is the least positive integer that is not a factor of not a factor of 25!. However, the question asks for an integer that is not a prime number, so 29 is not the answer.
At this point, you could consider 30, 31, 32, etc., but it is quicker to look at the rest of the choices. Choice C, 36, is equal to (4)(9). Both 4 and 9 are factors of not a factor of 25!, so 36 is also a factor of not a factor of 25!. Choice D, 56, is equal to (4)(14). Both 4 and 14 are factors of not a factor of 25!, so 56 is also a factor of not a factor of 25!. Choice E, 58, is equal to (2)(29). Although 2 is a factor of not a factor of 25!, the prime number 29, as noted earlier, is not a factor of not a factor of 25!, and therefore 58 is not a factor of not a factor of . The correct answer must be Choice E.
The explanation above uses a process of elimination to arrive at Choice E, which is sometimes the most efficient way to find the correct answer. However, one can also show directly that the correct answer is 58. For if a positive integer n is not a factor of not a factor of 25!, then one of the following must be true:
(i) n is a prime number greater than 25, like 29 or 31, or a multiple of such a prime number, like 58 or 62;
(ii) n is so great a multiple of some prime number less than 25, that it must be greater than 58.
To see that (i) or (ii) is true, recall that every integer greater than 1 has a unique prime factorization, and consider the prime factorization of not a factor of 25!. The prime factors of not a factor of are 2, 3, 5, 7, 11, 13, 17, 19, and 23, some of which occur more than once in the product not a factor of 25!. For example, there are 8 positive multiples of 3 less than 25, namely 3, 6, 9, 12, 15, 18, 21, and 24. The prime number 3 occurs once in each of these multiples, except for 9 and 18, in which it occurs twice. Thus, the factor 3 occurs 10 times in the prime factorization of not a factor of 25!. The same reasoning can be used to find the number of times that each of the prime factors occur, yielding the prime factorization. \(25!=(2^{22})(3^{10})(5^)(7^3)(11^2)(13)(17)(19)(23)\). Any integer whose prime factorization is a combination of one or more of the factors in the prime factorization of not a factor of 25!, perhaps with lesser exponents, is a factor of not a factor of 25!. Equivalently, if the positive integer n is not a factor of not a factor of 25!, then, restating (i) and (ii) above, the prime factorization of n must
(i) include a prime number greater than 25; or
(ii) have a greater exponent for one of the prime numbers in the prime factorization of not a factor of 25!.
For (ii), the least possibilities are \(2^{23}, 3^{11}, 5^7, 7^4, 11^3, 13^2, 17^2, 19^2, and 23^2\) . Clearly, all of these are greater than 58. The least possibility for (i) that is not a prime number is 58, and the least possibility for (ii) is greater than 58, so Choice E is the correct answer.
