If c and d are positives integers, d = 2 and c*d is the square of a

Question

If c and d are positives integers,  d\geq{2}  and c*d is the square of a positive integer, what is the least possible value of d?

(1) cd = 99,225
(2) c = 33,075

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GMATinsight · Answer

Question :  d_{min} = ?

Statement 1 : cd = 99,225 
 c*d = 9*11025 = 9*25*441 = 3^4*5^2*21^2 
So  d_{min} = 3  and  c = 3^3*5^2*21^2

SUFFICIENT

Statement 2 :  c = 33,075 = 25*9*7^2*3

for c^d to be a oerfect square we need atleast  3^1  to be multiplied with  25*9*7^2*3  which is c

hence d must be minimum = 3

SUFFICIENT

Answer: Option D

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • c and d positive integers.
• c*d are square of a positive integer.
 o Powers of all the prime factors of  c*d  will be even. 
•  d ≥ 2 
• We need to find the least possible value of d.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  cd = 99,225
 • From this statement we know the value of c*d.
 o Hence, we can do the prime factorization and see which is the smallest prime ≥ 2 in this prime factorization.
  This will be the value of d.

(• Prime factorization of  c*d = 25*3969 = 3*5^2*1323=5^2*3^2*441 =3^4*5^2*7^2 
• So, the least possible value of d is 3.)  
Hence, statement 1 is sufficient and we can eliminate answer options B, C and E
 Statement 2:  c = 33,075
 •From this statement, we can find prime factorization of c,then there will be two cases:
 o Case 1: c is perfect square, then smallest possible value of d will be  2^2  so that  c*d  is a perfect square.
o Case 2: If c is not a perfect square then the smallest prime ≥ 2 in the prime factorization of c will be d, so that c*d will be a perfect square. 
Thus, in either cases we can find the unique value of d.

(• c = 5^2*1323 =3*5^2*441 =3^3*5^2*7^2 
• Now, to make c*d a perfect square, the least value of d is 3.)  
Hence, statement 2 is also sufficient and the correct answer is  Option D. 
 NOTE :  This is a DS problem, so you can skip calculation and answer just by analysis.

satishametet · Answer

Statement 1 Example with Explanation

Statement 1:
c * d = 99,225 = 3^4 * 5^2 * 7^2

Possible values of d ≥ 2 that make c * d a perfect square:

d = 3 → c = 3^3 * 5^2 * 7^2
d = 7 → c = 3^4 * 5^2 * 7
d = 9 → c = 3^2 * 5^2 * 7^2
d = 21 → c = 3^4 * 5^2 * 7

Why it is not sufficient:
Even though c * d is a perfect square, Statement 1 alone does not tell us how the factors are split between c and d. Multiple valid values of d exist (3, 7, 9, 21), so we cannot uniquely determine the least possible d.

FrozenDragon · Answer

Hello Satish, here question asked the least "possible value" of c

So our goal here is to see whether statement A can answer that

As you correctly factored the number and found that the smallest possible number is 3, the question is answered.

That's why A is sufficient.

I hope this helps.

GulfTube · Answer

Given d >= 2, and cd is perfect square.

i) cd = 99225, prime factorize = 3^4∗5^2∗7^2, hence the least possible value of d is 3. This satisfies the condition given in question stem. Sufficient.

ii) c = 33075, prime factorize = 3^3∗5^2∗7^2.
 CONCEPT : Perfect square always has even power (exponent) for all of its prime factors.

Thus d must introduce odd exponent of 3 make the power of 3 even. Since question asks for least possible value of d, it is going to be 3^1. Sufficient.

Correct answer D

If c and d are positives integers, d >= 2 and c*d is the square of a

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