Gmat_mission wrote:What is the remainder when k² is divided by 8?
1) When k is divided by 2, the remainder is 1.
2) When k is divided by 3, the remainder is 2.
Target question: What is the remainder when k² is divided by 8?
Statement 1: When k is divided by 2, the remainder is 1.
This tells us that k is
one greater than some multiple of 2.
So, we can write k = 2n + 1, where n is some integer.
If k = 2n + 1, then k² = (2n + 1)² = 4n² + 4n + 1
So, the target question is basically asking us to determine the remainder when (4n² + 4n + 1) is divided by 8
To answer this question, I'll first show you that 4n² + 4n is divisible by 8
Notice that 4n² + 4n = 4n(n + 1) = 4(n)(n + 1)
Notice that n and n+1 are CONSECUTIVE integers, which means one of them is ODD and one is EVEN
Since one of them is EVEN, then we know that there's a 4 AND an EVEN number in the product 4(n)(n + 1)
So, the
4(n)(n + 1) must be divisible by 8
In other words,
4n² + 4n must be divisible by 8
From this, we can conclude that 4n² + 4n + 1 is
one greater than some multiple of 8
So, if we divide 4n² + 4n + 1 by 8, the remainder will be 1.
In other words,
if we divide k² by 8, the remainder will be 1
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: When k is divided by 3, the remainder is 2
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When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.
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So, from statement 2, we can conclude that some possible values of k are:
2, 5, 8, 11, 14, 17, . . . etc
Let's TEST two possible values of k
Case a: k = 2. In this case, k² = 2² = 4. When we divide 4 by 8, we get 0 with remainder 4. So,
the answer to the target question is 4
Case b: k = 5. In this case, k² = 5² = 25. When we divide 25 by 8, we get 3 with remainder 1. So,
the answer to the target question is 1
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
