Bunuel
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
Are You Up For the Challenge: 700 Level QuestionsSolution
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.