This one comes out of the MGMAT Quant section. Going through the solution of this problem, it appears that you can take the units digit of the individual components and manipulate algebraically, to get the units digit of the answer. In what other cases can you apply this rule? Thanks in advance.

Jared


QUESTION:
If (243)x(463)y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7

(2) x = 4

ANSWER:

We know from the question that x and y are integers and also that they are both greater than 0. Because we are only concerned with the units digit of n and because both bases end in 3 (243 and 463), we simply need to know x + y to figure out the units digit for n. Why? Because, to get the units digit, we are simply going to complete the operation 3x × 3y which, using our exponent rules, simplifies to 3(x + y).

So we can rephrase the question as "What is x + y?"

(1) SUFFICIENT: This tells us that x + y = 7. Therefore, the units digit of the expression in the question will be the same as the units digit of 37.

(2) INSUFFICIENT: This gives us no information about y.

The correct answer is A.

I should probably clarify the problem:

(243)^x (463)^y

The questem needs clarification. What you had stated above is a valid proposition

Great question. As long as you are dealing with multiplication (which also covers raising something to an exponent - multiplying something by itself), if you are asked for the units digit, you only need to deal with the units digits of whatever you have to manipulate. Think about the way multiplication works - whatever you multiply, only the units digits affect the units digit of the answer. You can ignore everything else. :)

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Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

I thought in a slightly different manner, does this question not asks what is value of n or is it sufficient to tell whether n can be found out; in my opinion since x + y can take following pair {(0,7),(1,6),(2,5),(3,4)} each resulting into different values of n; Thus both together would give correct answer ???

My choice was C but got it wrong, is my logic flawed please explain??

Since it's data sufficiency, the question whether we can determine the value of n, not what is the value of n. We just need to know the units digit, so we're only going to consider the units digits of the two numbers. Notice that with any of those pairs you listed above I have the following units digits:
(pair 0,7) 3^0 * 3^7
(pair 1,6) 3^1 * 3^6
(pair 2,5) 3^2 * 3^5
and so on.

Notice that since when multiplying the same bases (in this case, 3) we can add the exponents, and in this case x+y is always 7. So the units digit of this giant number will be the units digit of 3^7. Does this make sense?

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Jamie Nelson
ManhattanGMAT Instructor