Problem Solving

If y is divisible by the square of an even prime number and x is the actual square of an even prime number, then what is the units digit of xy?

0,2,4,6,8 are the answer choices

First, let's identify the value of the square of the only even prime number. The only even prime is 2, so the square of that is 22 = 4. Thus, x = 4 and y is divisible by 4. With this information, we know we will be raising 4 to some power divisible by 4. The next step is to see if we can establish a pattern.


41 = 4
42 = 16
43 = 64
44 = 256
45 = 1024

We will quickly notice that 4 raised to any odd power has a units digit of 4. And 4 raised to any even number has a units digit of 6. Therefore, because we are raising 4 to a number divisible by 4, which will be an even number, we know that the units digit of xy is 6.

The correct answer is D.


but y could also be 1, which will cause the digit number toi be 4...right?

hongwang9703 wrote:If y is divisible by the square of an even prime number and x is the actual square of an even prime number, then what is the units digit of xy?

0,2,4,6,8 are the answer choices

First, let's identify the value of the square of the only even prime number. The only even prime is 2, so the square of that is 22 = 4. Thus, x = 4 and y is divisible by 4. With this information, we know we will be raising 4 to some power divisible by 4. The next step is to see if we can establish a pattern.


41 = 4
42 = 16
43 = 64
44 = 256
45 = 1024

We will quickly notice that 4 raised to any odd power has a units digit of 4. And 4 raised to any even number has a units digit of 6. Therefore, because we are raising 4 to a number divisible by 4, which will be an even number, we know that the units digit of xy is 6.

The correct answer is D.


but y could also be 1, which will cause the digit number toi be 4...right?

As a Kaplan representative, I'd love to tell you that you're correct and that MGMAT is wrong; sadly, that's not the case.

Based on the explanation, the actual question is what's the units digit of x^y, not xy, as you posted.

As noted in the explanation, the only even prime is 2. Therefore, we know that x=4 and y is divisible by 4. Another way to say "y is divisible by 4" is "y is a multiple of 4".

If y is a multiple of 4, then y cannot equal 1, as you suggest. Y must be 4, 8, 12, 16, 20, ..., all of which fall in the same place in the pattern of units digits.

The only real problem I have with the question is that it doesn't say that y must be positive (did the question actually mention that?); if we let y=0, or any negative multiple of 4 (which we can do because 0 is divisible by 4, as is -4, -8, -12, ...), then 4^y won't have a units digit of 6.

So, if we're not restricted to positive values of y, there's no correct answer to the question. If, on the other hand, the original question read "If the positive number y is divisible by ...", then it's all good.

If y is divisible by the square of an even prime number and x is the actual square of an even prime number, then what is the units digit of x^y?

0
2
4
6
8

posted is the exact wording of the question, I am a bit confused, if Y is divisible By 4, couldnt Y be 1 because it is divisible by 4. 4/1 = 4

yes i do recall from a kaplan question that, 0 can be a divisible as well. such as 4/0 = 0, and 0 can be a multiple of any positive integer,such as 0 is a multiple of 4 ,is that correct?
sorry if the question seems very dumb but its just very confusing to me as i am trying to retain all these laws of numbers and formulas.

hongwang9703 wrote:If y is divisible by the square of an even prime number and x is the actual square of an even prime number, then what is the units digit of x^y?

0
2
4
6
8

posted is the exact wording of the question, I am a bit confused, if Y is divisible By 4, couldnt Y be 1 because it is divisible by 4. 4/1 = 4

yes i do recall from a kaplan question that, 0 can be a divisible as well. such as 4/0 = 0, and 0 can be a multiple of any positive integer,such as 0 is a multiple of 4 ,is that correct?
sorry if the question seems very dumb but its just very confusing to me as i am trying to retain all these laws of numbers and formulas.

You're not interpreting "divisible" correctly.

"Y is divisible by 4" means that y/4 is an integer; you've interpreted it as 4/y is an integer.

"Y is divisible by 4" means that "Y is a multiple of 4" - those are two ways of expressing the exact same relationship; further, it also means that "4 is a factor of Y".

No numbers are divisible by 0 (since anything divided by 0 is undefined); 0 is divisible by every number (except 0).

Since 0 is divisible by every number, 0 is a multiple of every number.

"Y is divisible by 4" means that "Y is a multiple of 4" - those are two ways of expressing the exact same relationship; further, it also means that "4 is a factor of Y".

ah hah! thats what I was doing wrong, interpretation! so y is divisible by 4 = 4/y = integer! hahleluah

Y is a multiple of 4 , = y=4x
here, X can be 1 or 0 which could make y=4 or 0 respectively??

ps, you are in Toroton which is only 2 hours away from me(Buffalo, NY) do you offer any private classes, if so how much do you charge.

thank you very much for taking ur time to answer my questions

[quote="hongwang9703ah hah! thats what I was doing wrong, interpretation! so y is divisible by 4 = 4/y = integer! hahleluah[/quote]

Still upside down!

Y is divisble by 4 = y/4 is an integer (not 4/y)

ps, you are in Toronto which is only 2 hours away from me(Buffalo, NY) do you offer any private classes, if so how much do you charge.

I'm exclusive to Kaplan - we do offer private tutoring (we have excellent tutors in Buffalo as well), which you can arrange either through our website (www.kaptest.com) or over the phone (1-800-KAP-TEST).