woohoo921 wrote:
GMATGuruNY wrote:
carcass wrote:
If m and n are positive integers, is m + n divisible by 4 ?
(1) m and n are each divisible by 2.
(2) Neither m nor n is divisible by 4.
Statement 1:
Case 1: m=2 and n=2, with the result that m+n=4
In this case, m+n is divisible by 4, so the answer to the question stem is YES.
Case 2: m=2 and n=4, with the result that m+n=6
In this case, m+n is not divisible by 4, so the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.
Statement 2:
Case 1: m=2 and n=2, with the result that m+n=4
In this case, m+n is divisible by 4, so the answer to the question stem is YES.
Case 3: m=1 and n=2, with the result that m+n=3
In this case, m+n is not divisible by 4, so the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.
Statements combined:
Since m is divisible by 2 but not by 4, m = 2a, where a is ODD.
Since n is divisible by 2 but not by 4, n = 2b, where b is ODD.
Thus:
m+n = 2a + 2b = 2(a+b) = 2(ODD + ODD) = 2(EVEN) = multiple of 4
The answer to the question stem is YES.
SUFFICIENT.
KarishmaBI am a bit confused on proving that the statements combined can be useful. How do we know to define a and b as both odd numbers? I would be so appreciative for your insights, and if you may have another way to efficiently arrive at the answer C.
Hi
woohoo921 Thanks for your query.
Let me give you another method to solve this problem in which I will not use plugging in as the main approach. Instead, I will show you the methodical approach to help you to arrive at the correct answer choice.
For this, first, let us analyze the question stem properly.
QUESTION STEM ANALYSIS: From the question stem, we only got that
m and n are POSITIVE INTEGERS. And for these
m and n, we need to find whether (m + n) is divisible by 4.
Let’s just first translate the entire question stem from English to Math. It has two parts- given and to find:
- Given: m and n belong to the set of positive integers {1, 2, 3, 4, …}
- To find: Whether (m + n) = 4k, for some positive integer k.
Note: You must always translate as you go. This will help you focus on the main workable elements of a question – the juice you extract from all its wordy sentences.
As this is all we can get from the question stem, let’s move to statement 1!
STATEMENT 1 ANALYSIS: “m and n are each divisible by 2.” This implies that
m and n can be represented as 2a and 2b, respectively, for positive integers ‘a’ and ‘b’. (Observe how we translated English to Math here as well, as our first task)
So, let’s check what we can now say about (m + n).
- m + n = 2a + 2b = 2(a + b).
- Since we finally need to check whether m + n = 4k or not, we must further analyze the nature of (a + b). There are two possibilities:
- If (a + b) is even: Then, (a + b) = 2k, (k is a positive integer), and hence, (m + n) = 2(2k) = 4k.
- So, here, the answer to the main question asked is YES!
- If (a + b) is odd: Then, (a + b) = 2k + 1, and hence, (m + n) = 2(2k + 1) = 4k + 2.
- Since 4k + 2 is not divisible by 4, the answer to our main question is NO!
As we are not getting a unique answer from statement 1 alone, it is
INSUFFICIENT.
Next, let’s move ahead and analyze statement 2.
STATEMENT 2 ANALYSIS: “
Neither m nor n is divisible by 4.”
In simple terms,
m ≠ 4p and
n ≠ 4q for any integers p and q, respectively.
So, m can take any form out of 4p + 1, 4p + 2, 4p + 3 but not 4p + 4 (this will again mean 4(p + 1) and hence, a multiple of 4). Similarly, n can take any form out of 4q + 1, 4q + 2, 4q + 3 but not 4q + 4.
Now, there can be many possible combinations of m and n that will give us different expressions for (m + n). Below are just two of those possibilities:
- If m = 4p + 1 and n = 4q + 1:
- m + n = 4(p + q) + 2 = 4(some integer) + 2. -> NOT a multiple of 4
- The answer to the main question asked is NO!
- If m = 4p + 2 and n = 4q + 2:
- m + n = 4(p + q) + 4 = 4(p + q + 1) = 4(some integer). -> a multiple of 4
- The answer to the main question asked is YES!
Again, we did not get a unique answer from statement 2 alone, making it
INSUFFICIENT. Since none of the statements alone comes out to be sufficient, let’s move ahead and analyze combined statements 1 and 2.
STATEMENT 1 AND 2 TOGETHER: From statement 1, we know that m = 2a and n = 2b, for integers a and b. And, from statement 2, we know that m ≠ 4p and n ≠ 4q, for any integers p and q.
So, m and n have
exactly one 2 as their factor. (If m and n had two 2s, they would also have a 4 which is not allowed per statement 2). Thus, we want such values of a and b, that
do not contribute another 2 to m and n.
Now, a and b can contribute another 2 to m and n only if they are
even. Since we
do not want this to happen, ‘a’ and ‘b’ are
NOT EVEN. That is, a and b are both
ODD.
Finally, if a and b are odd, we can express them as a = 2p + 1 and b = 2r + 1, for some integers p and r.
So, m + n = 2a + 2b can be simplified as:
- = 2(2p + 1) + 2(2r + 1) = 4p + 2 + 4r + 2
- = 4p + 4r + 4 = 4(p + r + 1)
- = 4k (divisible by 4)
And that’s it! The answer to our main question is a sure YES!
Thus, statements 1 and 2 combined are sufficient to answer our question.
Hence,
option C is the correct answer.
Hope this helps!
Best,
Aditi Gupta
Quant expert,
e-GMAT.
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