* If y is divisible by the square of...

[caplin]

I do not understand why the correct answer is D, why not C? Can you explain the sentence in bold below? Thank you

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?

[question edited/fixed by moderator]

A) 0
B) 2
C) 4
D) 6
E) 8

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.

4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024

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 x^y is 6.

[RonPurewal]

hi,
please read the forum rules before posting. you need to cite the original source of the problem.
if you don't cite the original source, we will have to delete this thread within a few days (to make sure we aren't running afoul of copyright laws/requests).

also, please be sure you've transcribed the problem correctly. as currently written, the "correct" answer to this problem could actually be any of the five given choices.
according to the info here, x = 4, and y is some multiple of 4 (i.e., y = 4 or 8 or 12 or 16 or ...)
so, xy could be
... 4 x 4 = 16 (answer d)
... 4 x 8 = 32 (answer b)
... 4 x 12 = 48 (answer e)
... 4 x 16 = 64 (answer c)
... 4 x 20 = 80 (answer a)
so, yeah, this problem probably isn't transcribed correctly.

--

within the next week, please
1/ transcribe the problem correctly
2/ cite the ORIGINAL SOURCE

thanks.

[caplin]

That question and explanation was copied and pasted directly from Manhattan GMAT CAT 3.

[RonPurewal]

That question and explanation was copied and pasted directly from Manhattan GMAT CAT 3.

are you sure it's not supposed to say x^y instead of xy?
if it's supposed to say x^y, then all of this will make a lot more sense.

please double-check and let us know, thanks.

[caplin]

Right, it should be X^Y -- formatting was lost when copying and pasting it in. I still do not understand why the answer is D instead of C.

[jlucero]

The sentence you are having trouble with:

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 x^y is 6.

Since x = 4 and y = 4, 8, 12, 16... or some other number divisible by 4, we know that 4^4 or 4^8 or 4^12... will all end in the number 6.

The pattern that they discovered was:
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024

Notice that when the exponent is an odd number, the units digit is 4 and when the exponent is an even number, the units digit is 6. Since y must be divisible by 4, it must be an even number. Since the units digit in x^y is even, than x^y must end in 6.

[caplin]

ok thank you. didn't realize that there was a connection with those even numbers ending in 6 being divisible by 4.

[jlucero]

No problem. Glad it makes sense now.

[caplin]

But 4^1, 4^3, and 4^5 are also each divisible by 4. So why couldnt the answer be C with a units digit of 4?

[tim]

The question is not about whether 4^y is divisible by 4. Notice that it stipulates that y itself must be divisible by 4, so 4^1, 4^3, 4^5, etc. are invalid.

[hzhang89]

2 isn't divisible by 4. This is important because y must be divisible by 4.

[jlucero]

2 isn't divisible by 4. This is important because y must be divisible by 4.

You are correct on the first part, but once you've eliminated all the odd values for y, any even value of y (whether divisible by 2 or by 4) will give you the same answer. So it might be important in some other question, but probably not this one.