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30 Dec 2021, 07:30
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
69% (02:19) correct
31%
(02:15)
wrong
based on 81
sessions
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A certain positive number has the property that ten times the number has a total of 35 factors. Which of the following could represent the ratio of the maximum value to the minmum value of the number?
(A) 4 : 25
(B) 16 : 25
(C) 25 : 32
(D) 25 : 4
(E) 4 : 125
(A) 4 : 25
(B) 16 : 25
(C) 25 : 32
(D) 25 : 4
(E) 4 : 125
06 Jan 2023, 10:33
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Bunuel
Let the number be x
We're given that 10*x has 35 factors
2 * 5 * x has 35 factors
35 can be expressed as 7 * 5, or 35 * 1
So the number, 2 * 5 * x, is in the form of \(p_1^6 * p_2^4\) or \(p^{34}\)
In this representation, \(p_1\), \(p_2\) and p are prime numbers.
As we already have two different prime numbers in the expression "2 * 5 * x" we can reject the possibility of the form \(p^{34}\) , therefore the only other possible form of 2*5*x is \(p_1^6 * p_2^4\). Hence we can infer that x is composed of 2s and 5s.
Min Value of x
To get the min value of x, we will minimize the power of 5 and maximize the power of 2
So the term 2 * 5 * x should be \(2^6 *5^4 \)
Therefore x must be \(2^5*5^3\)
Max Value of x
To get the max value of x, we will minimize the power of 2 and maximize the power of 5
So the term 2 * 5 * x should be \(5^6 *2^4 \)
Therefore x must be \(5^5*2^3\)
Maximum Value / Minimum Value =\(\frac{5^5*2^3 }{ 2^5*5^3}\) = \(\frac{5^2 }{ 2^2}\)
= \(\frac{25}{4}\)
Option D
Bhu750
Joined: 19 Nov 2020
Last visit: 12 Dec 2023
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Location: India
Schools: Fuqua '23 MIT MFin "22 Kellogg '23 Johnson '24 McDonough '24 Kellogg (S) LBS '25 (D) Emory 1YR '24 (WL)
GMAT 1: 700 Q48 V38
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Schools: Fuqua '23 MIT MFin "22 Kellogg '23 Johnson '24 McDonough '24 Kellogg (S) LBS '25 (D) Emory 1YR '24 (WL)
GMAT 1: 700 Q48 V38
Posts: 85
08 Jan 2023, 04:53
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35 can be obtained as 35*1 or 5*7
Therefore to get 35 as total factors there are 3 options -
Option 1 : \(a^{34}\).... Total factors = 34+1 = 35
Option 2 : \(a^{4}\) x \(b^{6}\).... Total factors = (4+1) x (6+1) = 35
Option 3 : \(a^{6}\) x \(b^{4}\).... Total factors = (4+1) x (6+1) = 35
We are given the total factors of 10 times a number is 35.. So there are 2 base factors which are 2 and 5.
Substitute 2 and 5 for a and b in Option 2 & 3 but reduce the power by 1 as we are given 10 times a number to get the maximum and minimum values
\(\frac{Maximum value}{Minimum value}\) = \(\frac{2^{3}*5^{5}}{2^{5}*5^{3}}\) = \(\frac{5^{2}}{2^{2}}\) = \(\frac{25}{4}\)
Answer is D \(\frac{25}{4}\)
Therefore to get 35 as total factors there are 3 options -
Option 1 : \(a^{34}\).... Total factors = 34+1 = 35
Option 2 : \(a^{4}\) x \(b^{6}\).... Total factors = (4+1) x (6+1) = 35
Option 3 : \(a^{6}\) x \(b^{4}\).... Total factors = (4+1) x (6+1) = 35
We are given the total factors of 10 times a number is 35.. So there are 2 base factors which are 2 and 5.
Substitute 2 and 5 for a and b in Option 2 & 3 but reduce the power by 1 as we are given 10 times a number to get the maximum and minimum values
\(\frac{Maximum value}{Minimum value}\) = \(\frac{2^{3}*5^{5}}{2^{5}*5^{3}}\) = \(\frac{5^{2}}{2^{2}}\) = \(\frac{25}{4}\)
Answer is D \(\frac{25}{4}\)
