What is the value of the positive integer m ?

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.

General rule for reminder: y=divisor*quotient+reminder=xq+r, where 0 <equal r < x

(1) m=6q+3 so m can be 3, 9, 15, etc. not sufficient.

(2) 15=mq+6. So mq=9. We have 3 possibilities:
1: m=1 and q=9
2: m=3 and q=3
3: m=9 and q=1
But as from the general rule (0 <equal r < x) m>6, m=9
Sufficient

Answer B.

Here we go -


St1: When m is divided by 6, the remainder is 3.
m = 6a + 3
m can take multiple values..
Clearly insufficient


St2: When 15 is divided by m, the remainder is 6.

15 = mq + 6 ---> mq = 9

as m is a positive integer

(m,q) = (1,9)
(m,q) = (3,3)
(m,q) = (9,1)

Remainder = 6 (given)

0<= remainder < divisor(m)

Only 9 satisfies.

Clearly B is sufficient.

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.

Statement 1. M=6q+3 Hence m could be 3,9,12,15 and so on. Insufficient
Statement 2. (15)=mq+6. Insufficient since we don't know the values of m and q
Both statements together. mq=9 ==>q=(9/m) and hence m=(54/m)+3 ==> we can multiply both part by m and get an equation (m^2)-3m-54=0. This equation has two roots 9 and (-6). Since m is a positive integer (-6) is out and we are left with m=9. Hence sufficient and answer C. But i am sure that there is a faster way to find a solution
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15/m = q + 6

m thus has to be more than 6
15 = mq +6
mq = 9
m can be only 9, since m has to be an integer, and q too.
B.

Excellent Question
Here the concept we need is => Remainder≥0 and Remainder<divisor
in statement 1 => M=3,9etc=> Not suff
in sattement 2 => M must be greater than 6 so M=7,8,9..... but 15/M must generate a remainder of 5 => hence M=9 only

Smash that B
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Question: m=?

(1) m/6 --> R=3

Therefore m could be 3,9,15....

NOT SUFFICIENT

(2) 15/m --> R=6

There is only one number, 9, that would allow us to get a remainder of 6. We know m must be positive, therefore we are safe to say the number is 9.

SUFFICIENT

B.

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

RELATED VIDEO FROM OUR COURSE

Asked: What is the value of the positive integer m ?

(1) When m is divided by 6, the remainder is 3.
m = 6k + 3; where k is an integer
m = {3,9,15,21,....}
There are mulitple values of m possible
NOT SUFFICIENT

(2) When 15 is divided by m, the remainder is 6.
15 = mn + 6
mn = 9
m = 9/n
m = {9,3,1}: But 15 is completely divisible by 3 & 1.
m = 9 is the only possibility
SUFFICIENT

IMO B

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