Nups1324 wrote:
smyarga wrote:
So, my solution. Just a little bit different from the previous.
The remainder when \(N\) is divided by 18 is 16 means that \(N=18q+16\) for some integer \(q\).
\(N\) is a multiple of 28 means that \(N=28s\) for some integer \(s\).
We need to find the remainder when \(\frac{N}{4}\) is divided by 18.
On one hand \(\frac{N}{4}=7s\), on the other hand \(\frac{N}{4}=\frac{9q}{2}+4\). Since \(7s=\frac{9q}{2}+4\) and \(s\) is an integer, \(q\) must be even.
So, \(\frac{N}{4}=9k+4\) for some integer \(k\).
If\(k\) is even (\(k=2n\) for some integer \(n\)) the remainder when \(\frac{N}{4}\) is divided by 18 is 4 (\(\frac{N}{4}=9*2n+4=18n+4\)).
If \(k\) is odd (\(k=2n+1\) for some integer \(n\)) the remainder when \(\frac{N}{4}\) is divided by 18 is 13 (\(\frac{N}{4}=9(2n+1)+4=18n+13\)).
So, there two possible values for the remainder 4 and 13.
The answer is B.
Hi
smyargaI did not understand this step and thus nothing after it:
So, \(\frac{N}{4}=9k+4\) for some integer \(k\).
How did the 9q/2 changed into 9k?
Tagging others just in case
Bunuel chetan2u GMATinsight ScottTargetTestPrep IanStewart yashikaaggarwalThank you
Posted from my mobile deviceWe know N/4 = 9q/2 + 4 and we know q is even. Since q is even, we can express q as q = 2k for some integer k. Substituting q by 2k, we obtain:
N/4 = 9(2k)/2 + 4 = 9k + 4
That's how the expression 9k + 4 is obtained. Now, the integer k is either even or odd. If k is even, k can be expressed as k = 2n for some integer n and if k is odd, k can be expressed as k = 2m + 1 for some integer m. In one case, we obtain the equality N = 18n + 4 and in the other case, we obtain N = 18m + 13.
At this point, we need to recall the definition of division with remainders: when a is divided by b, we say that the quotient is q and the remainder is r if a = bq + r and 0 ≤ r < b.
Using the equality N = 18n + 4, we see that r = 4 satisfies 0 ≤ 4 < 18; so that's the remainder when N is divided by 18. Similarly, the equality N = 18m + 13 tells us that the remainder when N is divided by 18 is 13. Thus, there are two possible values for the remainder when N is divided by 18.
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