A certain positive number has the property that ten times the number

Question

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

gmatophobia · Answer

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 · Answer

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}

RiyaJain69 · Answer

I got the answer without doing any maths. The question asks for the ratio of maximum: minimum. Obviously, the first value will be bigger of the two. In this question, only D fits the bill. (Yes, if the question were to have multiple options with bigger:smaller number, then I'd have to do the maths.)

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A certain positive number has the property that ten times the number

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