lheiannie07 wrote:What is the remainder when x^4 + y^4 divided by 5?
(1) x - y divided by 5 gives remainder 1
(2) x + y divided by 5 gives remainder 2
Can some experts explain how can i figure the correct statement?
OA C
The question must mention that x and y are integers, otherwise the answer is E.
We have to determine the remainder when x^4 + y^4 divided by 5.
(1) x - y divided by 5 gives remainder 1.
Case 1: Say x - y = 6, and x = 6, and y = 0. We see that x^4 + y^4 = 6^4 + 0^4 = 6^4; We see that the remainder when 6^4 = 256 divided by 5 is 1.
Case 2: Say x - y = 6, and x = 7, and y = 1. We see that x^4 + y^4 = 7^4 + 1^4; We see that the remainder when 7^4 divided by 5 is 1 and the remainder when 1^4 divided by 5 is also 1. Thus, the remainder when x^4 + y^4 = 7^4 + 1^4 divided by 5 is 1 + 1 = 2. No unique answer. Insufficient.
(2) x + y divided by 5 gives remainder 2
Case 1: Say x + y = 7, and x = 7, and y = 0. We see that x^4 + y^4 = 7^4 + 0^4 = 7^4; We see that the remainder when 7^4 (The unit digit of 7^4 is 1) divided by 5 is 1.
Case 2: Say x + y = 7, and x = 3, and y = 4. We see that x^4 + y^4 = 3^4 + 4^4; We see that the remainder when 3^4 divided by 5 is 1 and the remainder when 4^4 = 256 divided by 5 is also 1. Thus, the remainder when x^4 + y^4 = 3^4 + 4^4 divided by 5 is 1 + 1 = 2. No unique answer. Insufficient.
(1) and (2) together
Say x - y = 5p + 1 and x + y = 5q + 1
Thus, x = 5(p + q)/2 + 3/2 and y = 5(q - p)/2 + 1/2
Since (p + q)/2 and (q - p)/2 are the quotients, let's replace them with a and b. Thus. x = 5a + 3/2 and y = 5b + 1/2
Thus, x^4 + y^4 = (5a + 3/2)^4 + (5b + 1/2)^4
In the expansion of (5a + 3/2)^4 + (5b + 1/2)^4, all the terms except (3/2)^4 and (1/2)^4 will have at least one multiple of 5, thus those terms are divisible by 5, leaving 0 as remainder.
Thus, the remainder when x^4 + y^4 = (5a + 3/2)^4 + (5b + 1/2)^4 divided by 5 = (3/2)^4 + (1/2)^4 divided by 5 = 81/16 + 1/16 divided by 5 = 82/16 divided by 5 = 2. Sufficient,
The correct answer:
C
Hope this helps!
-Jay
_________________
Manhattan Review GMAT Prep
Locations:
New York |
Singapore |
Doha |
Lausanne | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor!
Click here.
