If n is a positive integer, and (45*14*7^n - 15*7^(n - 1)*54)^(1/2) is

Question

If n is a positive integer, and  \sqrt{45*14*7^n - 15*7^{(n - 1)}*54}  is a positive integer, what is the value of n?

(1) n is a factor of a prime number
(2) n < 3

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M36-53

CareerGeek · Answer

\sqrt{45*14*7^n - 15*7^{(n - 1)}*54} 
=  \sqrt{2*3^2*5*7^{n + 1} - 2*3^4*5*7^{(n - 1)}} 
=  \sqrt{2*3^2*5*7^2*7^{n - 1} - 2*3^4*5*7^{(n - 1)}} 
=  \sqrt{2*3^2*5*7^{n - 1}(7^2 - 3^2)} 
=  \sqrt{2*3^2*5*7^{n - 1}(40)} 
=  \sqrt{2*3^2*5*7^{n - 1}(2^3*5)} 
=  \sqrt{2^4*3^2*5^2*7^{n - 1}} 
=  2^2*3*5\sqrt{7^{n - 1}}  
=  60\sqrt{7^{n - 1}}  is a positive integer

The above is a positive integer for all odd values of n = {1, 3, 5, 7,, 9, . . . . . . }

(1)  n  is a factor of a prime number
--> Possible values of  n = {1, 2, 3, 5, 7, 9, . . . . . .}  --> Not always Odd -->  Insufficient

(2)  n < 3 
--> Possible values of  n = 1  -->  Sufficient

Option B

Kinshook · Answer

\sqrt{45*14*7^n - 15*7^{(n - 1)}*54} = \sqrt{3^2*5*2*7*7^n - 3*5*7^{(n - 1)}*2*3^3} =   
  \sqrt{2*3^2*5*7^{n+1} - 2*3^4*5*7^{(n - 1)}} = \sqrt{2*3^2*5*7^2*7^{n-1} - 2*3^2*5*3^2*7^{(n - 1)}} =   
  \sqrt{2*3^2*5*7^{n-1} (49 - 9)} = \sqrt{2*3^2*5*7^{n-1} 2^3*5} =   
  \sqrt{2^4*3^2*5^2*7^{n-1}} = 4*3*5 * \sqrt{7^{n-1}}

For  \sqrt{7^{n-1}}  to be an integer (n-1) must be an even number or n must be an odd number

(1) n is a factor of a prime number
n = 1 or a prime number
If n=2; n-1 is odd ; NOT FEASIBLE
For all other prime numbers n-1 is even; FEASIBLE
NOT SUFFICIENT

(2) n < 3
n= 1 or 2
If n=1; n-1=0 ; FEASIBLE
But if n=2; n-1 is odd; NOT FEASIBLE
NOT SUFFICIENT

(1) + (2) 
(1) n is a factor of a prime number
(2) n < 3
n = 1 or 2
If n=1; n-1=0 ; FEASIBLE
But if n=2; n-1 is odd; NOT FEASIBLE
NOT SUFFICIENT

IMO E

Bunuel · Answer

Similar question: https://gmatclub.com/forum/if-n-is-a-po ... 18660.html P.S. Anyone else wants to try above question?

eakabuah · Answer

√(45x14x7^n - 15x7^(n-1) x 54) = √(15x3x2x7^(n+1) - 15x3x2x9x7^(n-1)) = √(15x6x7^(n-1) x (7^2 - 9))
=√(90x7^(n-1) x 40) = √(3600x7^(n-1)) = 60√(7^(n-1)
The question is asking what value of n makes √(7^(n-1) an integer?

Statement 1: n is a factor of a prime number
possible values of n: 1,2,3,5,7,13, 17
when n is 1, √(7^(1-1) = 1, which is an integer.
when n=3, √(7^(3-1) = 7, which is an integer.
Statement 1 is insufficient.

Statement 2: n < 3
possible values of n: 1,2
when n=1 √(7^(1-1) = 1 an integer.
when n=2 √(7^(2-1) = √7 not an integer, hence we can conclude that n=1
Statement 2 is sufficient.

The answer is B.

Bunuel · Answer

Official Solution:

If  n  is a positive integer, and  \sqrt{45*14*7^n - 15*7^{(n - 1)}*54}  is an integer, what is the value of  n ? 
 
Simplify given expression:
 
 \sqrt{45*14*7^n - 15*7^{(n - 1)}*54}=  
 
 =\sqrt{2*3^2*5*7^2*7^{(n-1)} - 2*3^4*5*7^{(n - 1)}}=  
 
 =\sqrt{2*3^2*5*7^{(n-1)}(7^2 - 3^2)}=  
 
 =\sqrt{2*3^2*5*7^{(n-1)}(2^3*5)}=  
 
 =\sqrt{2^4*3^2*5^2*7^{(n-1)}}=  
 
 =2^2*3*5*\sqrt{7^{(n-1)}}  
 
So, we are given that  2^2*3*5\sqrt{7^{(n-1)}}  is an integer. 
 
Notice that since  n  is a positive integer, then  7^{(n-1)}  is also a positive integer.
 
Now,  the square root of any positive integer is either an integer or an irrational number.  Meaning that,  \sqrt{positive \ integer}  cannot be a fraction, for example it cannot equal to  \frac{1}{2}, \ \frac{3}{7}, \ \frac{19}{2}, \ \frac{1}{60}  ... It MUST be an integer (1, 2, 3, ...) or an irrational number (for example  \sqrt{2} ,  \sqrt{3} ,  \sqrt{7} , ...). Thus, since  7^{(n-1)}  is a  positive integer  then  \sqrt{7^{(n-1)}}  is either an integer itself or an irrational number.
 
Therefore, for  2^2*3*5*\sqrt{7^{(n-1)}}  to be an integer,  \sqrt{7^{(n-1)}}  must be an integer.  \sqrt{7^{(n-1)}}  is an integer for (positive) ODD values of  n . 
 
So, from above,  n  is a positive odd integer: 1, 3, 5, 7, ... 
 
(1)  n  is a factor of a prime number
 
 n  can be 1 or any odd prime. Not sufficient.
 
(2)  n < 3 
 
There is only one positive odd number less than 3, namely 1. So,  n = 1 . Sufficient.

Answer: B

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If n is a positive integer, and (45147^n - 157^(n - 1)54)^(1/2) is

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