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Difficulty:
95%
(hard)
Question Stats:
33% (02:06) correct
67%
(01:55)
wrong
based on 226
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How many prime factors does \(x^{37}\) have?
(1) The number of prime factors of 16x is one more than the number of prime factors of \(x^4\)
(2) \(2x^{16}\) has 2 prime factors
(1) The number of prime factors of 16x is one more than the number of prime factors of \(x^4\)
(2) \(2x^{16}\) has 2 prime factors
09 Mar 2015, 13:52
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How many prime factors does \(x^{37}\) have?
Real GMAT question would mention that x is a positive integer. ALL GMAT divisiblity questions are limited to positive integers. So, I'd assume that it's given that x is a positive integer.
First of all note that exponentiation does not "produce" primes, meaning that for positive integers m and n, m^n would have as many prime factors as m itself does.
(1) The number of prime factors of 16x is one more than the number of prime factors of x^4 --> 16x = 2^4*x. So it's given that 2^4*x has one more prime than x^4, which means that 2 is NOT a prime factor of x. We know nothing else about x. Not sufficient.
(2) \(2x^{16}\) has 2 prime factors --> this implies that x itself can have 1 prime (something other than 2) or 2 primes (2 and something other than 2). Not sufficient.
(1)+(2) Since from (1) we know that 2 is NOT a factor of x, then from (2) it follows that x must have only 1 prime. Thus x^37 also will have only 1 prime. Sufficient.
Answer: C.
Hope it's clear.
Real GMAT question would mention that x is a positive integer. ALL GMAT divisiblity questions are limited to positive integers. So, I'd assume that it's given that x is a positive integer.
First of all note that exponentiation does not "produce" primes, meaning that for positive integers m and n, m^n would have as many prime factors as m itself does.
(1) The number of prime factors of 16x is one more than the number of prime factors of x^4 --> 16x = 2^4*x. So it's given that 2^4*x has one more prime than x^4, which means that 2 is NOT a prime factor of x. We know nothing else about x. Not sufficient.
(2) \(2x^{16}\) has 2 prime factors --> this implies that x itself can have 1 prime (something other than 2) or 2 primes (2 and something other than 2). Not sufficient.
(1)+(2) Since from (1) we know that 2 is NOT a factor of x, then from (2) it follows that x must have only 1 prime. Thus x^37 also will have only 1 prime. Sufficient.
Answer: C.
Hope it's clear.
General Discussion
09 Mar 2015, 08:19
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TARGET730
ans C....
1) statement 1 tells us that 16x has one more prime factor than \(x^4\), which will be same as for x.... therefore it tells us that x is not multiple of 2... insufficient...
2) statement 2 tells us that \(2x^{16}\) has two prime factor .... it tells us that that one prime factor is 2 along with one more prime factor...
does not tell us that x also has a 2 in it... insufficient...
combined it tells us that x has only one prime factor that is itself.... sufficient
