If x is a positive integer less than 100 such that x is divisible by 2y, where y is a positive integer, what is the value of y?a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'B'. Can you explain this answer?

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If x is a positive integer less than 100 such that x is divisible by 2^y, where y is a positive integer, what is the value of y?

a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Correct answer is option 'B'. Can you explain this answer?

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If x is a positive integer less than 100 such that x is divisible by 2...

Steps 1 & 2: Understand Question and Draw Inferences

x is an integer such that 0 < x < 100
x is divisible by 2^y, where y is a positive integer
Since y > 0, this means 2 is definitely a prime factor 2
Since x is a positive integer, we can write the Prime-factorized form of x as:
are integers > 0, z ≥ y and are prime numbers other than 2.
- As x < 100, 2^z < 100
- So, z = { 1, 2, 3, 4, 5, 6} as 2⁷ = 128 > 100
- Since x is completely divisible by 2^y, z ≥ y. So, y can take any value of z, i.e. 1 ≤ y ≤ 6
To Find: Unique value of y

Step 3: Analyze Statement 1 independently

As x is a positive integer, x < -60 is not possible.
So, x > 60, i.e. 60 < x < 100.

However, we do not know if 2 is the only prime factor or x. So, we cannot find a unique value of y.

For example, if 2 is the only prime factor of x, then y can have only 1 value: 6
But if x has other prime factors, then multiple values of y are possible. For example, x could be 2²*3^*7 (y = 2) or 2³*11 (y = 3) etc.

Insufficient to answer.

Step 4: Analyze Statement 2 independently

Using the prime factorized expression of x, we can write:
Since the integer resulting from this division is odd, the powers of 2 in the numerator and the denominator should cancel out each other.
Hence 2^2z = 2^y+2
z=y/2+1 ….(1). So, y must be even as z is an integer
Also, from our discussion in Steps 1 and 2, we know that z ≥ y
Substituting (1) in the above inequality, we get:

Using (2), along with the inference that y must be even, we have y = 2 as the only possible option.

Sufficient to answer.

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step 4, this step is not required.

Answer: B

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If x is a positive integer less than 100 such that x is divisible by 2y, where y is a positive integer, what is the value of y?a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'B'. Can you explain this answer?

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