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If x, y are positive integers, what is the remainder when 2^

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If x, y are positive integers, what is the remainder when 2^

by Max@Math Revolution » Mon Jul 04, 2016 10:45 pm
If x, y are positive integers, what is the remainder when 2^(8x+y) is divided by 5?
1) x=1
2) y=2

*An answer will be posted in 2 days
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by 800_or_bust » Tue Jul 05, 2016 6:10 am
Max@Math Revolution wrote:If x, y are positive integers, what is the remainder when 2^(8x+y) is divided by 5?
1) x=1
2) y=2

*An answer will be posted in 2 days
To start, I decided to look for a pattern when dividing powers of 2 by 5.

Notice: 2^0/1 has a remainder of 1, 2^1/5 has a remainder of 2, 2^2/5 has a remainder of 4, 2^3/5 has a remainder of 3, 2^4/5 has a remainder of 1, 2^5/5 has a remainder of 2, 2^6/5 has a remainder of 4, 2^7/5 has a remainder of 3, and 2^8/5 has a remainder of 1. By now, a pattern has clearly emerged. When 2 is raised to a power that is a multiple of 4, the remainder is 1. And from there it cycles from 1 to 2 to 4 to 3, and then back to 1 (with the next power that's a multiple of 4).

(1) Not sufficient. We have a power equal to 8 plus an unknown positive integer. The remainder could be 1,2,3 or 4, depending on the value of y.

(2) Sufficient. We have a multiple of 8 to which we are adding 2. Every multiple of 8 will yield a remainder of 1, and, since we are adding 2 to the multiple of 8, the remainder will be 4.

Answer: B
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by Max@Math Revolution » Wed Jul 06, 2016 3:48 pm
From the original condition, there are 2 variables (x and y). In order to match the number of variables and the number of equations, we need 2 equations. Hence, there is a high chance that C is the correct answer. However, even though C is indeed the correct answer, since it is an integer question, one of the key questions, we have to apply the common mistake type 4(A). From the condition 2), the remainder of 2^(8x+y) is always 4. Hence, it is sufficient and the correct answer is B.

- Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.
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