If y is an integer, is y^2 divisible by 4?

St1: y is even - Zero is also even so it could be 0 in this case 2,4 in any of those cases it's divisible by 4 SUFFICIENT
St2: y is an even integer in this case because if it were odd odd*odd*odd is odd and not divisible by 4 - SUFFICIEINT

Answer (D)

VERITAS PREP OFFICIAL SOLUTION:

Question Type: Yes/No. This question asks: “Is y^2 divisible by 4.”

Given information in the question stem or diagram: y is an integer.

Statement 1: y is even. With a conceptual understanding of factors, you know that if y is an even number then y^2 will have to be a multiple of 4. If you want to quickly test this you can use some small even numbers. For example, if y = 2, then y^2=4; yes, that is divisible by 4. If y = 4, then y^2 = 16; this is also divisible by 4. Even 0, which is an even number, will work. 0^2 = 0; and yes, 0 is divisible by 4. This confirms what we already knew from our conceptual understanding: an even
number when squared is a multiple of 4. The answer is either A or D.

Statement 2: y^3 is divisible by 4. For some, this may not seem sufficient at first glance. y^3 might be divisible by 4 while y^2 is not. For example, y could be the cube root of 4. However, if you leverage properly the information in the question stem (always so important!) then you know that “y is an integer.” So y cannot be the cube root of 4; it must be an integer. Once you have established that y must be an integer then the only way for y^3 to be divisible by 4 is for y to be an even number. In other words, it gives you the same information that you already found sufficient in Statement 1. Thus, the correct answer is D.

Note: This is another good example of “When one statement is easy, the other is hard/counterintuitive.” The first statement is relatively simple so you should be extra careful analyzing the second statement. As is often the case, the key with Statement 2 is to make sure you properly leverage the given information.

Solution:
Pre Analysis:

• y is an integer
• We are asked if \(y^2\) is divisible by 4 or not

Statement 1: y is even

• Since y is even, we can assume \(y=2k\) where k is a non-negative integer
• So, \(y^2=(2k)^2=4k^2\)
• This means \(y^2\) is a multiple of 4
• Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: \(y^3\) is divisible by 4

• According to this statement, \(y^3=4k\) or \(y^3=2^2k\)
• This means \(y^2\) at least has to be \(2^2\) or a multiple of 4
• Thus, statement 2 alone is also sufficient

Hence the right answer is Option D