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A positive integer x has 60 divisors and 7x has 80 divisors. What is t

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A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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22 Jan 2015, 02:01
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A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4
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Location: Pune, India
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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22 Jan 2015, 02:43
3
1
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

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Karishma
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Current Student Status: The Final Countdown Joined: 07 Mar 2013 Posts: 288 Concentration: Technology, General Management GMAT 1: 710 Q47 V41 GPA: 3.84 WE: Information Technology (Computer Software) Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink] Show Tags 10 Feb 2015, 11:51 I am still unable to understand this explanation.. Total number of factors of x = 60 = 2^2 * 3 * 5(Its fine till here) = (3+1)*(2+1)*(4+1)(why this??) Total number of factors of 7x = 80 = 2^2 * 4 * 5(Its fine till here) = (3+1)*(3+1)*(4+1)(why this??) And this part leaves me The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2. Could you please elaborate on this?How can a 3 become a 4?and why?and how does 7 come into the picture? TIA! Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8102 Location: Pune, India Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink] Show Tags 10 Feb 2015, 22:38 1 Ralphcuisak wrote: I am still unable to understand this explanation.. Total number of factors of x = 60 = 2^2 * 3 * 5(Its fine till here) = (3+1)*(2+1)*(4+1)(why this??) Total number of factors of 7x = 80 = 2^2 * 4 * 5(Its fine till here) = (3+1)*(3+1)*(4+1)(why this??) And this part leaves me The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2. Could you please elaborate on this?How can a 3 become a 4?and why?and how does 7 come into the picture? TIA! Tia, I think you need to check out this post first: http://www.veritasprep.com/blog/2010/12 ... ly-number/ This tells you that if N is of the form a^p*b^q*c^r..., the Total Number of Factors of a number N = (p+1)(q+1)(r+1)... (note that this is the number of factors of N, not N itself) Here you are given that total number of factors on x is 60. Remember, x is not 60. The total number of factors of x is 60. You need to write 60 in the form (p+1)(q+1)(r+1)... One way of doing that is $$60 = 4*3*5 = (3+1)*(2+1)*(4+1)$$ Note that 4 = 3+1, 3 = 2+1, 5 = 4+1. So we have done nothing except changed the form. When will we write that total number of factors of x are (3+1)*(2+1)*(4+1)? This happens when x is of the form $$a^3 * b^2 * c^4$$ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Posts: 5911
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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11 Feb 2015, 21:27
3
1
VeritasPrepKarishma wrote:
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

hi,
you have given a good way of solving it .. kudos for that

i can think of one straight way to do it..
let the number(x..n..?) be a^k*b^l...7^t..
here we are just interested in value of 't'.
two points now..

1) n=a^k*b^l...7^t.. so number of factors=(k+1)(l+1)..(t+1)=60..
lets take all other values as z ie z=(k+1)(l+1).....
so z(t+1)=60....(1)

2)7n=a^k*b^l...7^(1+t).. so number of factors=(k+1)(l+1)..(t+2)=80..
as we take all other values as z ie z=(k+1)(l+1).....
so z(t+2)=80....(2)

from eq (1)and(2)...
80(t+1)=60(t+2)... t=2..
so ans is 2.. C
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Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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11 Feb 2015, 21:42
chetan2u wrote:
VeritasPrepKarishma wrote:
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

hi,
you have given a good way of solving it .. kudos for that

i can think of one straight way to do it..
let the number(x..n..?) be a^k*b^l...7^t..
here we are just interested in value of 't'.
two points now..

1) n=a^k*b^l...7^t.. so number of factors=(k+1)(l+1)..(t+1)=60..
lets take all other values as z ie z=(k+1)(l+1).....
so z(t+1)=60....(1)

2)7n=a^k*b^l...7^(1+t).. so number of factors=(k+1)(l+1)..(t+2)=80..
as we take all other values as z ie z=(k+1)(l+1).....
so z(t+2)=80....(2)

from eq (1)and(2)...
80(t+1)=60(t+2)... t=2..
so ans is 2.. C

hi chetan2u,
super bro.. how could simplify it so easily ..
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Status: Math is psycho-logical
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Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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02 Mar 2015, 14:10
VeritasPrepKarishma wrote:
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

One question I have refers to the difference of x and 7x. What do we mean with "7x has only one 7 more than x" and with "Only the power of 7 increases by 1"?

I do understand the difference between (3+1)*(2+1)*(4+1) and (3+1)*(3+1)*(4+1), and I can see that the 2 became a 3. But I don't know how this relates to 7...
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Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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02 Mar 2015, 20:04
1
pacifist85 wrote:
One question I have refers to the difference of x and 7x. What do we mean with "7x has only one 7 more than x" and with "Only the power of 7 increases by 1"?

I do understand the difference between (3+1)*(2+1)*(4+1) and (3+1)*(3+1)*(4+1), and I can see that the 2 became a 3. But I don't know how this relates to 7...

Ask yourself, why does 2 become 3?

Say $$x = 2^3 * 7^2 * 11^4$$
Total number of factors of $$x = (3+1)*(2+1)*(4+1) = 60$$

What will be 7x?

$$7x = 2^3 * 7^3 * 11^4$$
When you prime factorize 7x, you will get another 7.
Total number of factors of $$7x = (3+1)*(3+1)*(4+1) = 80$$
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Status: Math is psycho-logical Joined: 07 Apr 2014 Posts: 423 Location: Netherlands GMAT Date: 02-11-2015 WE: Psychology and Counseling (Other) Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink] Show Tags 03 Mar 2015, 01:52 VeritasPrepKarishma wrote: pacifist85 wrote: One question I have refers to the difference of x and 7x. What do we mean with "7x has only one 7 more than x" and with "Only the power of 7 increases by 1"? I do understand the difference between (3+1)*(2+1)*(4+1) and (3+1)*(3+1)*(4+1), and I can see that the 2 became a 3. But I don't know how this relates to 7... Ask yourself, why does 2 become 3? Say $$x = 2^3 * 7^2 * 11^4$$ Total number of factors of $$x = (3+1)*(2+1)*(4+1) = 60$$ What will be 7x? $$7x = 2^3 * 7^3 * 11^4$$ When you prime factorize 7x, you will get another 7. Total number of factors of $$7x = (3+1)*(3+1)*(4+1) = 80$$ So, the point it that you can get whatever number using this number of factors for x: (3+1)*(2+1)*(4+1). Then, you see than when x becomes 7x, the number of factors of one of the bases increases: (3+1)*(3+1)*(4+1). Since the only thing that changed is that 7x has an additional prime factor (7), we conclude that this 7 belonged to (2+1) that became (3+1). Then we know that it was 2 before and this is how we get to our answer. Manager Joined: 14 Jul 2014 Posts: 187 Location: United States Schools: Duke '20 (D) GMAT 1: 600 Q48 V27 GMAT 2: 720 Q50 V37 GPA: 3.2 Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink] Show Tags 11 Apr 2016, 09:08 How is n related to x? I didn't understand this question at all.. Senior Manager Joined: 03 Apr 2013 Posts: 288 Location: India Concentration: Marketing, Finance Schools: Simon '20 GMAT 1: 740 Q50 V41 GPA: 3 A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink] Show Tags 31 Oct 2016, 04:53 manpreetsingh86 wrote: A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n? a) 0 b) 1 c) 2 d) 3 e) 4 Here's another method to do this. I have used algebraic method here.. First of all..notice that a power of 7 has to be a part of n already for the given condition to be true, because if that is not the case..then a completely new 7 will make the number of factors as 120(exactly double). The number of factors of n can be written as (a+1)(b+1)(c+1)...(s+1)(t+1) = 60...(t+1) is for powers of 7 ------ eq(1) Let (a+1)(b+1)...(s+1) = k According to the next condition.. (a+1)(b+1)....(s+1)(t+2) = 80 Or k(t+2) = 80 => k(t+1+1) = 80 => k(t+1) + k = 80 Using the given value => 60 + k = 80 => k = 20 Using this value in eq(1) (t+1) = 3 => t = 2 And Voila! Answer(C) _________________ Spread some love..Like = +1 Kudos Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8102 Location: Pune, India Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink] Show Tags 31 Oct 2016, 05:42 dina98 wrote: How is n related to x? I didn't understand this question at all.. Note that n is a typo. The question is "What is the greatest integer y such that 7^y divides x? _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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31 Oct 2016, 10:06
As I understand there should be x instead of n.
If x has 60 divisors we can represent it as follows:

1*60; 2*30; 3*20; 4*15 ; 5*12 ; 6*10 ; 2*2*15 ; 2*3*10 ; 2*2*3*5
$$p^{59}$$ ; $$p*q^{29}$$ ; $$p^2*q^{19}$$ ; $$p^3*q^{14}$$ ; $$p^4*q^{11}$$ ... (etc. - power of prime is 1 less than its representation in factor grouping)

Now if 7 is not present in initial number x, then in 7*x we’ll have 120 as total number of factors (we are adding 2 additional choices for additional prime and due to multiplication rule the whole result will be doubled).

So the only possibility is that 7 is already a prime factor of x. In this case, we just need to increase its power by 1. The obvious choice we can spot at first glance is 3*20. In order to get 80 we need 4*20. So the initial power of 7 in x was one less than 3.

Kind of amateurish by I hope understandable
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A positive integer x has 60 divisors and 7x has 80 divisors. What is t [#permalink]

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01 Jan 2018, 21:15
I solved it like this:

First thing to note: if x does not have 7 as one of the prime factors, then 7x will have double the factors i.e. 120

so x already has 7 as one of the prime factors.
say prime factorization of $$x = (p + 1)(q+1)(r+1).... = 60$$ (total factors)
Let p be the power of 7 in x

then prime factorization of $$7x = (p+1+1)(q+1)(r+1).... = 80$$
substituting $$(q+1)(r+1)....$$ as = $$60/(p+1)$$
=> $$(p+2) * 60/(p+1) = 80$$
=> $$3p + 6 = 4p + 4$$ => $$p = 2$$
so x has $$7^2$$ has the factor.

So maximum y such that $$7^y$$ will divide x is 2 => (C)
A positive integer x has 60 divisors and 7x has 80 divisors. What is t   [#permalink] 01 Jan 2018, 21:15
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