Bunuel wrote:
If x and y are positive integers and n = 5^x + 7^(y + 3), what is the units digit of n?
(1) y = 2x – 16
(2) y is divisible by 4.
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:The units digit of n is determined solely by the units digit of the expressions 5^x and 7^(y + 3), because when two numbers are added together, the units digit of the sum is determined solely by the units digits of the added numbers.
Since x is a positive integer, and 5^(any positive integer) always has a units digit of 5, 5^x always ends in a 5.
However, the units digit of 7^(y + 3) is not certain, as the units digit pattern for the powers of 7 is a four-term repeat: [7, 9, 3, 1].
The question can thus be rephrased as “what is the units digit of 7^(y + 3)?”
Note: Determining y would be one way of answering the question above, but we should not rephrase to “what is y?” Because the units digits of the powers of 7 have a repeating pattern, we might get a single answer for the units digit of 7^(y + 3) despite having multiple values for y.
(1) INSUFFICIENT: This statement tells us neither the value of y nor the units digit of 7^(y + 3), as y depends on the value of x, which could be any positive integer. For example, if x = 9, then y = 2 and 7^(y + 3) has a units digit of 7. By contrast, if x = 10, then y = 4 and 7^(y + 3) has a units digit of 3.
(2) SUFFICIENT: Regardless of what multiple of 4 we pick, 7^(y + 3) will have the same units digit. Ultimately this means that n has a units digit of 5 + 3 = 8.
The correct answer is B. _________________