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# If x is a positive integer, what is the number of different positive

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Joined: 02 Sep 2009
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If x is a positive integer, what is the number of different positive  [#permalink]

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16 Dec 2019, 01:40
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55% (hard)

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50% (01:12) correct 50% (01:04) wrong based on 26 sessions

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If x is a positive integer, what is the number of different positive factors of 39x ?

(1) x is a two-digit number
(2) x^2 has 3 positive factors

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If x is a positive integer, what is the number of different positive  [#permalink]

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Updated on: 16 Dec 2019, 06:46
Bunuel wrote:
If x is a positive integer, what is the number of different positive factors of 39x ?

(1) x is a two-digit number
(2) x^2 has 3 positive factors

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(1) x is a two-digit number

If x is 11 (or any 2 digit prime), the number of factors of $$39x=3^1*11^1*13^1$$ is $$(1+1)*(1+1)*(1+1)=8$$
If x is 12, the number of factors of $$39x=2^2*3^2*13$$ is $$(2+1)*(2+1)*(1+1)=18$$

1 is not sufficient

(2) x^2 has 3 positive factors

Note that only squares of prime numbers can have 3 positive factors

So x is a prime number and so the number of factors for $$39x=3^1*13^1*x^1$$ is $$(1+1)*(1+1)*(1+1)=8$$

But if x=13, the number of factors is (1+1)*(2+1)=6

2 is not Sufficient

(1)+(2)

x can still be 13 or any other 2 digit prime

Not sufficient

Originally posted by firas92 on 16 Dec 2019, 03:58.
Last edited by firas92 on 16 Dec 2019, 06:46, edited 1 time in total.
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Re: If x is a positive integer, what is the number of different positive  [#permalink]

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16 Dec 2019, 06:26
Top Contributor
Bunuel wrote:
If x is a positive integer, what is the number of different positive factors of 39x ?

(1) x is a two-digit number
(2) x² has 3 positive factors

------ASIDE---------------
Key concept: If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
---------------------------
Given: x is a positive integer

Target question: What is the number of different positive factors of 39x ?

Statement 1: x is a two-digit number
Let's TEST some values of x:
Case a: x = 11. So, 39x = (3)(13)(11) = (3¹)(13¹)(11¹). So, the number of positive divisors = (1+1)(1+1)(1+1) = 8. The answer to the target question is 39x has 8 positive divisors
Case b: x = 13. So, 39x = (3)(13)(13) = (3¹)(13²). So, the number of positive divisors = (1+1)(2+1) = 6. The answer to the target question is 39x has 6 positive divisors
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x² has 3 positive factors
This tells us that x must be a prime number
IMPORTANT: Notice that we can use the same x-values we used to show that statement 1 is not sufficient:
Case a: x = 11. So, 39x = (3)(13)(11) = (3¹)(13¹)(11¹). So, the number of positive divisors = (1+1)(1+1)(1+1) = 8. The answer to the target question is 39x has 8 positive divisors
Case b: x = 13. So, 39x = (3)(13)(13) = (3¹)(13²). So, the number of positive divisors = (1+1)(2+1) = 6. The answer to the target question is 39x has 6 positive divisors
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Since I was able to use the same counter-examples to show that each statement ALONE is not sufficient, the same counter-examples will satisfy the two statements COMBINED.
Case a: x = 11. So, 39x = (3)(13)(11) = (3¹)(13¹)(11¹). So, the number of positive divisors = (1+1)(1+1)(1+1) = 8. The answer to the target question is 39x has 8 positive divisors
Case b: x = 13. So, 39x = (3)(13)(13) = (3¹)(13²). So, the number of positive divisors = (1+1)(2+1) = 6. The answer to the target question is 39x has 6 positive divisors
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Re: If x is a positive integer, what is the number of different positive   [#permalink] 16 Dec 2019, 06:26
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