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09 Jun 2015, 07:49
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D
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Difficulty:
25%
(medium)
Question Stats:
73% (01:08) correct
27%
(01:06)
wrong
based on 205
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What is the least positive integer that is not a factor of 30! and is not a prime number?
A. 31
B. 32
C. 33
D. 62
E. 64
A. 31
B. 32
C. 33
D. 62
E. 64
09 Jun 2015, 07:50
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Official Solution:
What is the least positive integer that is not a factor of 30! and is not a prime number?
A. 31
B. 32
C. 33
D. 62
E. 64
We are looking for a number that is not a factor of 30! and is also not a prime number.
The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is 2*31=62. Notice that all numbers between 30 and 62 are either primes (and we know that x is NOT a prime) or factors of 30! (because 30! has all its primes in higher powers). For example:
31 is a prime, hence \(x\) cannot be 31.
\(32 = 2^5\). Since 30! surely has 2 raised to a power greater than 5, 32 IS a factor of 30!.
\(33 = 3*11\). Both 3 and 11 are factors of 30!, hence 33 IS a factor of 30!
...
\(64 = 2^6\). Since 30! surely has 2 raised to a power greater than 6, 64 IS a factor of 30!.
Answer: D
What is the least positive integer that is not a factor of 30! and is not a prime number?
A. 31
B. 32
C. 33
D. 62
E. 64
We are looking for a number that is not a factor of 30! and is also not a prime number.
The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is 2*31=62. Notice that all numbers between 30 and 62 are either primes (and we know that x is NOT a prime) or factors of 30! (because 30! has all its primes in higher powers). For example:
31 is a prime, hence \(x\) cannot be 31.
\(32 = 2^5\). Since 30! surely has 2 raised to a power greater than 5, 32 IS a factor of 30!.
\(33 = 3*11\). Both 3 and 11 are factors of 30!, hence 33 IS a factor of 30!
...
\(64 = 2^6\). Since 30! surely has 2 raised to a power greater than 6, 64 IS a factor of 30!.
Answer: D
30 Aug 2023, 01:40
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
