As is typical with the high level
E-GMAT questions, we have a larger question stem (great question 😁 )
X has to be a + integer < 100
Y has to be a + integer
And
X / (2)^y = Integer
What is the value of y = ?
S1: (X)^2 > 3,600
Since we are given that X is a positive integer, we do not have to worry about the (-)negative root when we take the square root both sides:
[X] > sqrt(3,600)
[X] > 60
since X > 0 ——-> [X] = X
X > 60
X can take the value of 64 = (2)^6
We just need to meet the constraint that:
64 / (2)^y
y could = 1, 2, 3, 4, etc. —— not a unique value for Y
S1 NOT sufficient
S2:
(X)^2 / (2)^(y + 2) = Odd Integer
Concept: since X must be a positive integer, anytime we square X it must have an Identical row of prime factors because (X)^2 = X * X —— thus the Powers of all the Prime Bases that make up (X)^2 will be EVEN Powers
Further, in order to have an Odd Integer Quotient result after diving (X)^2 by (2)^(y +2) ————>
the value of (X)^2 must include/cancel out all the Prime Factors of 2 in the DEN As well as having all of the Primes in the NUM cancel
Because of these two facts, Y can not take an Odd Value (inference 1)
For example, if Y took an Odd value:
Case 1: let Y = 1
(X)^2 / (2)^3 = Odd Integer
In order to have an Odd Integer quotient, two things must happen:
(1st) all the 2 prime factors in the DEN must cancel
And
(2nd) there must be no prime factors of 2 left in the NUM
If we let X = 2 * 2 * 3 for example, when we square X this will double the Prime Factors and result in:
(X)^2 = (2)^4 * (3)^2
When we divide by (2)^3, the DEN will cancel out and we will be left with the following as the quotient:
(2) * (3)^2 = 18 ——- which is NOT an Odd Integer
Therefore, y must take an Even Positive Integer Value:
Case 2: let Y = 2
(X)^2 / (2)^4 = ODD Int.
We can let X = (2)^2 * 3
When we square X it will double all the prime factors and we will have:
(X)^2 = (2)^4 * (3)^2
When we divide by (2)^4 ———> both the Primes of 2 in the NUM and DEN will cancel out and we will be left with a Quotient = 9 = Odd integer
Furthermore, we must satisfy the Given Constraint in which X = (2)^2 * 3 is divisible by (2)^y
In this case, since y = 2, we will meetthe constraint as:
(2)^2 * 3 is divisible by (2)^2
Thus: Y = 2 is a possible value
If we try to make Y any larger, we cannot have a Value of X that is both:
-less than 100
-satisfies statement 2 when squared
And
-divisible by (2)^y
Try case 3: let y = 4
(X)^2 / (2)^6 = Odd Integer
We need to make sure that the primes in X are just enough so that they cancel our (2)^6, otherwise we will have an Even Int. Quotient.
In other words, (X)^2 must contain no more than (2)^6 in its prime factorization.
Let X = (2)^3 * (3)
This will satisfy statement 2 once we square X ———> (X)^2 = (2)^6 (3)^2
And when we divide by (2)^6 ——-> quotient = 9 = Odd Integer. So we can satisfy statement 2.
BUT unfortunately the Constraint in the question stem will be violated if Y = 4:
X / (2)^y = Int.
(2)^3 * (3) / (2)^4 ————> not an integer
Again, we can not add any more (2) primes to X, because then we will not satisfy statement 2 ——— adding anymore (2) primes to X and they will not be canceled out by (2)^6 and we will be left with an Even Integer Quotient
As you move up and try higher and higher Even Values for Y, the same problem will keep happening. We can meet statement 2’a criteria, but we can not simultaneously satisfy the given constraint.
Therefore, under the constraint and statement 2, the only value that Y can take is:
Y = 2
B - s2 is sufficient alone
Hard question
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