For positive integers m and n, when n is divided by 7, the quotient is m and the remainder is 2. What is the remainder

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For positive integers m and n, when n is divided by 7, the quotient is m and the remainder is 2. What is the remainder when m is divided by 11?

(1) When n is divided by 11, the remainder is 2.
(2) When m is divided by 13, the remainder is 0.



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Source: Veritas Prep

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BTGmoderatorDC wrote:
Sat Feb 05, 2022 4:34 pm
For positive integers m and n, when n is divided by 7, the quotient is m and the remainder is 2. What is the remainder when m is divided by 11?

(1) When n is divided by 11, the remainder is 2.
(2) When m is divided by 13, the remainder is 0.



OA A

Source: Veritas Prep
Target question: What is the remainder when m is divided by 11?

Given: When n is divided by 7, the quotient is m and the remainder is 2
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, with the given information, we can write: n = 7m + 2

Statement 1: When n is divided by 11, the remainder is 2.
In other words, n divided by 11 equals some unstated integer (say k) with remainder 2.
Applying the above rule, we can write: n = 11k + 2 (where k is some integer)
Since we already know that n = 7m + 2, we write the following equation:
7m + 2 = 11k + 2
Subtract 2 from both sides to get: 7m = 11k
Divide both sides by 7 to get: m = 11k/7
Or we can say m = (11)(k/7)
What does this tell us?
First, it tells us that, since m is an integer, it MUST be true that k is divisible by 7.
It also tells us that m is divisible by 11
If m is divisible by 11, then when m is divided by 11, the remainder will be 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When m is divided by 13, the remainder is 0
Applying the above rule, we can write: m = 13j (where j is some integer)
We already know that n = 7m + 2, but that doesn't help us much this time.
We COULD take n = 7m + 2 and replace m with 13j to get n = 7(13j) + 2. However, this doesn't get us very far, since the target question is all about what happens when we divide m by 11, and our new equation doesn't even include m.
At this point, I suggest that we start TESTING VALUES.
There are several values of m that satisfy statement 2. Here are two:
Case a: m = 13, in which case m divided by 11 gives us a remainder of 2
Case b: m = 26, in which case m divided by 11 gives us a remainder of 4
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

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BTGmoderatorDC wrote:
Sat Feb 05, 2022 4:34 pm
For positive integers m and n, when n is divided by 7, the quotient is m and the remainder is 2. What is the remainder when m is divided by 11?

(1) When n is divided by 11, the remainder is 2.
(2) When m is divided by 13, the remainder is 0.



OA A

Source: Veritas Prep
\(n=7m+2\)

From 1,
\(n=11k+2\, \Longrightarrow \, 7m+2=11k+2\, \Longrightarrow \, 7m=11k.\) We have \((7,11)=1,\) so \(m\) is divided by \(11.\) The remainder is \(0.\) Sufficient \(\Large{\color{green}\checkmark}\)

From 2,
\(m=13k,\) so \(m\) could be \(13, 26, 39, \ldots\) and the remainder when \(m\) is divided by \(11\) could be \(2, 4, 6, \ldots \) Not Sufficient \(\Large{\color{red}\chi}\)

Therefore, A