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a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 02:50
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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18 May 2015, 06:10
Bunuel wrote: a = 5^15  625^3 and a/x is an integer, where x is a positive integer such that it does NOT have a factor p such that 1 < p < x, then how many different values for x are possible?
A. None B. One C. Two D. Three E. Four
Kudos for a correct solution. OFFICIAL SOLUTION:First of all, notice that x is a positive integer such that it does NOT have a factor p such that 1 < p < x simply means that x is a prime number. Next, \(a = 5^{15}  625^3=5^{15}  5^{12}=5^{12}(5^31)=5^{12}*124=2^2*5^{12}*31\). Finally, for a/x to be an integer where x is a prime, x can take 3 values: 2, 5, or 31. Answer: D.
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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Updated on: 15 May 2015, 10:09
Bunuel wrote: a = 5^15  625^3 and a/x is an integer, where x is a positive integer such that it does NOT have a factor p such that 2 < p < x, then how many different values for x are possible?
A. None B. One C. Two D. Three E. Four
Kudos for a correct solution. a= 5^15  625^3 => 5^15  (5^4)^3 => 5^15  5^12 = 5^12(5^3  1) = 5^12*124 124 = 31*4 a/x is integer For condition of 2<p<x only 5 and 31 satisfies this Hence answer is C
Originally posted by King407 on 15 May 2015, 09:31.
Last edited by King407 on 15 May 2015, 10:09, edited 1 time in total.



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 09:49
a=5^15625^3=5^155^12=5^12*124 Now a/x=integer and 124=4x31 Only number satisfying the given conditions are 5 and 31 C
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 10:04
Resolve a= (625^3)*(625^4)625^3 = 625^3(625^41); so it can have two values one is 625^3 & (625^41) => multiple of 5 and 4
Given that a=+ int * X, therefore X can definitely have two values.
But it may have more than two as well, as it can be 5 * 25*625…*4. So here I’m confused. But still I will go with answer two, option C
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 11:52
This is a tricky worded question and I think the answer is should be D not C... Here is my reason : The stem says that x is a positive integer such that has no factor grater than 2 and less than x itself . The stem wants to say that X is a PRIME NUMBER . because any prime Number has no factor grater than 1 and Itself . On the other hand the stem says that X COULD get how many different number NOT MUST get different number ( this is very important issue ) AS our friends say, if we simplify Numerator more we can obtain : 5^12 ( 5^31) = 5^12 (124) = 5^12 (31*2*2) divided by x and we are told that this fraction is An integer . so, X COULD Be ( not Must be) 5 , 31 ,or 2 !!! so , X could get 3 different values and answer is D.... Best Regards,



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 12:00
mehrdadtaheri92 wrote: This is a tricky worded question and I think the answer is should be D not C... Here is my reason : The stem says that x is a positive integer such that has no factor grater than 2 and less than x itself . The stem wants to say that X is a PRIME NUMBER . because any prime Number has no factor grater than 1 and Itself . On the other hand the stem says that X COULD get how many different number NOT MUST get different number ( this is very important issue ) AS our friends say, if we simplify Numerator more we can obtain : 5^12 ( 5^31) = 5^12 (124) = 5^12 (31*2*2) divided by x and we are told that this fraction is An integer . so, X COULD Be ( not Must be) 5 , 31 ,or 2 !!! so , X could get 3 different values and answer is D.... Best Regards, Here we have the inequality 2<p<x and I assumed that x >2 is a given hence ignored 2. Otherwise we need not consider x >2 and can include 1 to the list and that would make 4 roots.
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 12:07
masoomdon wrote: mehrdadtaheri92 wrote: This is a tricky worded question and I think the answer is should be D not C... Here is my reason : The stem says that x is a positive integer such that has no factor grater than 2 and less than x itself . The stem wants to say that X is a PRIME NUMBER . because any prime Number has no factor grater than 1 and Itself . On the other hand the stem says that X COULD get how many different number NOT MUST get different number ( this is very important issue ) AS our friends say, if we simplify Numerator more we can obtain : 5^12 ( 5^31) = 5^12 (124) = 5^12 (31*2*2) divided by x and we are told that this fraction is An integer . so, X COULD Be ( not Must be) 5 , 31 ,or 2 !!! so , X could get 3 different values and answer is D.... Best Regards, Here we have the inequality 2<p<x and I assumed that x >2 is a given hence ignored 2. Otherwise we need not consider x >2 and can include 1 to the list and that would make 4 roots.[/quote X can not be 1 since from the stem we should understand that X is a prime number and 1 is NOT a prime number . so 1 CAN NOT include in the possible roots,...



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 12:13
Its actually not mentioned that x is a prime number . from the inequality 2<p<x we assume that x has to be greater than 2 for the equality to make sense but if we forgo that then both 2 and 1 will be on the roots. I still feel only numbers will be 5 and 31.
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 12:22
masoomdon wrote: Its actually not mentioned that x is a prime number . from the inequality 2<p<x we assume that x has to be greater than 2 for the equality to make sense but if we forgo that then both 2 and 1 will be on the roots. I still feel only numbers will be 5 and 31. OK . May be you right. let wait till the OE and OA. But if you read the question stem more carefully, you will see that it says X has NO factor GRATER than 2 and less THAN x itself. I think it means X is a prime number. Because any prime number has ONLY 1 and itself in its factor list...



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 12:29
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For a prime number the inequality would be 2<=p<=x whereas here we have 2<p<x.anyways let's wait for the OA and OE I remembered it just now so edited the post instead of adding one more post and hijacking the thread
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 15:28
the answer is E the question say that x does not has factore less than x itself and more than 2 which mean that x does not have factors in this range a =(5^155^12) =5^12(5^31)=5^12*124 so x could be 1, 5,2,13 all these numbers do not have factores less than the numbers itself and in the same time these factors are more than 2In other word, the two condations sould not be find to say that the number is xfirst condition is that the number has factors less than the number itself secondly the factors should be more than 2. for example 2 have the factore 1 which is less than 2 itself this is the first does not match with x condition were x does not havs this point but 1 is not more than 2 so we can say that 2 could be x were 2 DOES NOT has a factor in this range. finally we can not say that x should be prime number because if x is prime number then x will be DO NOT HAS FACTOR 1<P<X.
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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15 May 2015, 20:16
Bunuel wrote: a = 5^15  625^3 and a/x is an integer, where x is a positive integer such that it does NOT have a factor p such that 2 < p < x, then how many different values for x are possible?
A. None B. One C. Two D. Three E. Four
Ans: D Solution: a=5^12*31*4 (after generalizing it) now a/x is an integer x must be a factor. so the question can have three different factors, which satisfy the condition 2< p <x are 5, 31, 4
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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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16 May 2015, 00:22
e is the answer x could be 2,4,31 & 5



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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18 May 2015, 06:11



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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19 May 2015, 07:29
I just wonder why x can't be 1. "1" does not violate the statement that "x is a positive integer such that it does NOT have a factor p such that 1 < p < x".



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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19 May 2015, 07:33



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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19 May 2015, 07:45
Bunuel wrote: VladimirKarpov wrote: I just wonder why x can't be 1. "1" does not violate the statement that "x is a positive integer such that it does NOT have a factor p such that 1 < p < x". 1 < p < x means that 1 < x. But the statement gives us information that scheme "1<p<x" does not work here. I don't think we can interpret it as 1 < x. I will make substitution in the statement: "1 is a positive integer such that it does NOT have a factor p such that 1<p<x". That's right  it does not have.



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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19 May 2015, 07:49



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Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer
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03 Dec 2016, 07:43
a = 5^15  5^12 (As 625 = 5^4 and 5^4^3 = 5^12) a= 5^12(5^3  1) a = 5^12(124)...124 = 2*2*31
So, 2, 5, 31 are prime numbers (Not having divisors other than 1 and self)
So, D is the correct answer




Re: a = 5^15  625^3 and a/x is an integer, where x is a positive integer &nbs
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03 Dec 2016, 07:43



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