Is a positive integer k is divisible by 3?

Question

(1)  \sqrt{k}  is divisible by 9

(2) k^2 is divisible by 3

M36-19

Bunuel · Accepted Answer

Official Solution:

Is a positive integer  k  is divisible by 3? 
 
(1)  \sqrt{k}  is divisible by 9
 
 \sqrt{k}=9x , for a positive integer  x . Square:  k = 81x^2= a \ multiple \ of \ 3 . Sufficient.
 
(2)  k^2  is divisible by 3
 
Exponentiation does not "produce" primes (meaning that if  m  and  n  are positive integers, then  m^n  will have the same prime factors as  m  itself). So, if 3 is a factor of  k^2 , then 3 must also be a factor of  k . Sufficient.

Answer: D

DavidTutorexamPAL · Answer

The answer is D. 
 
1) This means that k is divisible by 81 ==> it is definitely divisible by 3! 
2) this is only true for square numbers (9, 36, 81) whose roots (3, 6, 9) are divisible by 3 themselves. Yes!

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

The only catch in this question is to remember that k is a positive integer such as 1 or 2 or 3 or so on...

Let's look at the statements:

Statement 1: If  \sqrt{k}  is divisible by 9, naturally k will be divisible by 3 also since  3^2 =9

Statement 2: Now, here's where the catch comes into play. If  k^2  is divisible by 3 and k is a positive integer, k will definitely be divisible by 3 too. Note here, that we cannot take k to be a value such as  \sqrt{3} , this is because k is a positive integer.

So, clearly, both statements are sufficient and  answer is D

TestPrepUnlimited · Answer

Statement 1:

We can write this as  \sqrt{k} = 9*i , where  i  is a nonnegative integer. Then we have  k = 81*i^2 , hence k is a multiple of 81. Then k must be divisible by 3. Sufficient.

Statement 2:

We are given k is already an integer. If  k^2  is divisible by 3, the factor of 3 must come from a single k so k must have a factor of 3. Sufficient.
(Another way to think about this is we fill in the square,  k^2  must be divisible by 9 given it is divisible by 3).

Ans: D

Is a positive integer k is divisible by 3?

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GMAT Data Sufficiency (DS) Questions

Is a positive integer k is divisible by 3?

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