What is the value of the positive integer \(m?\)

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What is the value of the positive integer \(m?\)

(1) When \(m\) is divided by \(6,\) the remainder is \(3.\)
(2) When \(15\) is divided by \(m,\) the remainder is \(6.\)

Answer: B

Source: GMAT Prep

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Gmat_mission wrote:
Sun Jan 23, 2022 1:22 am
What is the value of the positive integer \(m?\)

(1) When \(m\) is divided by \(6,\) the remainder is \(3.\)
(2) When \(15\) is divided by \(m,\) the remainder is \(6.\)

Answer: B

Source: GMAT Prep
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.

Target question: What is the value of positive integer m?

Statement 1: When m is divided by 6, the remainder is 3
According to the above rule, we can write the following:
The possible values of m are: 3, 3+6, 3+(2)(6), 3+(3)(6)...
Evaluate to get: the possible values of m = 3, 9, 15, 21, etc.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT


Statement 2: When 15 is divided by m, the remainder is 6
According to the above rule, we can conclude that....
Possible values of 15 are: 6, 6 + m, 6 + 2m, 6 + 3m, ...

Aside: Yes, it seems weird to say "possible values of 15," but it fits with the language of the above rule]

Now, let's test some possibilities:
15 = 6...nope
15 = 6 + m. Solve to get m = 9. So, this is one possible value of m.
15 = 6 + 2m. Solve to get m = 4.5
STOP. There are 2 reasons why m cannot equal 4.5. First, we're told that m is a positive INTEGER. Second, the remainder (6 in this case) CANNOT be greater than the divisor (4.5)

If we keep going, we get: 15 = 6 + 3m. Solve to get m = 3. Here, m cannot equal 3 because the remainder (6) CANNOT be greater than the divisor (3).
If we keep checking possible values (e.g., 15 = 6 + 3m, 15 = 6 + 4m, etc), we'll find that all possible values of m will be less than the remainder (6).

So, the ONLY possible scenario here is that m must equal 9
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B
Brent Hanneson - Creator of GMATPrepNow.com
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