If the positive integer n is greater than 6, what is the remainder when n is divided by 6?
(1) When n is divided by 9, the remainder is 2.
(2) When n is divided by 4, the remainder is 1.
The OA is the option C.
Are sufficient both statements? I need some help here. Experts, can you show me how to solve this DS question? Thanks in advanced.
If the positive integer n is greater than 6, what is
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-A quick lesson on remainders:
In other words, n is 2 more than a MULTIPLE OF 9:
n = 9a + 2, where a is nonnegative integer.
Options for n:
2, 11, 20, 29...
Since n must be greater than 6:
If n=11, then dividing by 6 yields a remainder of 1.
If n=20, then dividing by 6 yields a remainder of 2.
Since the remainder can be different values, INSUFFICIENT.
Statement 2:
In other words, n is 1 more than a MULTIPLE OF 4:
n = 4b + 1, where b is nonnegative integer.
Options for n:
1, 5, 9, 13, 17, 21, 25, 29.,,
Since n must be greater than 6:
If n=9, then dividing by 6 yields a remainder of 3.
If n=13, then dividing by 6 yields a remainder of 1.
Since the remainder can be different values, INSUFFICIENT.
Statements combined:
When n is divided by 36 -- the LCM of 9 and 4 -- the remainder will be 29 (the smallest value common to the two lists in red):
n = 36c + 29 = 29, 65, 101...
When the values in the blue list are divided by 6, the remainder in every case is 5.
SUFFICIENT.
The correct answer is C.
Onto the problem at hand:When x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3.
When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4.
Combined, the statements above imply that when x is divided by 35 -- the LOWEST COMMON MULTIPLE OF 5 AND 7 -- there will be a constant remainder R.
Put another way, x is R more than a multiple of 35:
x = 35c + R.
To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13, 18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11, 18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.
Putting it all together:
x = 35c + 18.
Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10, 13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2, 13...
Thus, when x is divided by 33 -- the LCM of 3 and 11 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...
Statement 1:M7MBA wrote:If the positive integer n is greater than 6, what is the remainder when n is divided by 6?
(1) When n is divided by 9, the remainder is 2.
(2) When n is divided by 4, the remainder is 1.
In other words, n is 2 more than a MULTIPLE OF 9:
n = 9a + 2, where a is nonnegative integer.
Options for n:
2, 11, 20, 29...
Since n must be greater than 6:
If n=11, then dividing by 6 yields a remainder of 1.
If n=20, then dividing by 6 yields a remainder of 2.
Since the remainder can be different values, INSUFFICIENT.
Statement 2:
In other words, n is 1 more than a MULTIPLE OF 4:
n = 4b + 1, where b is nonnegative integer.
Options for n:
1, 5, 9, 13, 17, 21, 25, 29.,,
Since n must be greater than 6:
If n=9, then dividing by 6 yields a remainder of 3.
If n=13, then dividing by 6 yields a remainder of 1.
Since the remainder can be different values, INSUFFICIENT.
Statements combined:
When n is divided by 36 -- the LCM of 9 and 4 -- the remainder will be 29 (the smallest value common to the two lists in red):
n = 36c + 29 = 29, 65, 101...
When the values in the blue list are divided by 6, the remainder in every case is 5.
SUFFICIENT.
The correct answer is C.
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Here's another (longer!) approach (MItch's solution is much nicer, but this one will also work)M7MBA wrote:If the positive integer n is greater than 6, what is the remainder when n is divided by 6?
(1) When n is divided by 9, the remainder is 2.
(2) When n is divided by 4, the remainder is 1.
Target question: What is the remainder when n is divided by 6?
Given: Positive integer n is greater than 6
Statement 1: When n is divided by 9, the remainder is 2.
------ASIDE----------------
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.
------------------------------
So, from statement 1, we can conclude that some possible values of n are: 11, 20, 29, 38, 47, 56, 65. . . (since we're told n > 6, we can rule out n = 2) .
Let's test some possible values of n:
Case a: n = 11. When we divide 11 by 6, the remainder is 5
Case b: n = 20. When we divide 20 by 6, the remainder is 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: When n is divided by 4, the remainder is 1.
from statement 2, we can conclude that some possible values of n are: 9, 13, 17, 21, 25, 29, 33, 37, 41, . . . (since we're told n > 6, we can rule out n = 1 and n = 5) .
Let's test some possible values of n:
Case a: n = 9. When we divide 9 by 6, the remainder is 3
Case b: n = 13. When we divide 13 by 6, the remainder is 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that some possible values of n are: 11, 20, 29, 38, 47, 56, 65, . . .
Statement 2 tells us that some possible values of n are: 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69 . . .
From the above, we can see that some values of n that satisfy BOTH statements are: 29, 65, etc
We might also notice that if we keep adding 36 (the least common multiple of 9 and 4) to these possible n-values, we'll get more possible n-values such as 101, 137, 173, etc
Let's test some possible values of n:
Case a: n = 29. When we divide 29 by 6, the remainder is 5
Case b: n = 65. When we divide 65 by 6, the remainder is 5
Case c: n = 101. When we divide 101 by 6, the remainder is 5
Case d: n = 137. When we divide 137 by 6, the remainder is 5
So, it certainly looks like the remainder will always be 5.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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We are given that n > 6 and we need to determine the remainder when n is divided by 6.M7MBA wrote:If the positive integer n is greater than 6, what is the remainder when n is divided by 6?
(1) When n is divided by 9, the remainder is 2.
(2) When n is divided by 4, the remainder is 1.
Statement One Alone:
When n is divided by 9, the remainder is 2.
So n can be 11, 20, 29, ...
11/6 = 1 remainder 5
20/6 = 3 remainder 2
Since there could be at least two different remainders, statement one alone is not sufficient to answer the question.
Statement Two Alone:
When n is divided by 4, the remainder is 1.
So n can be 9, 13, 17, 21, 25, 29, ...
9/6 = 1 remainder 3
13/6 = 2 remainder 1
Since there could be at least two different remainders, statement two alone is not sufficient to answer the question.
Statements One and Two Together:
From the list of values of n that have a remainder of 2 when divided by 9 and the list of value of n that have a remainder of 1 when divided by 4, we see that the first value of n can be 29. By keep adding the LCM of 9 and 4, i.e., 36, we will generate a list of values of n that satisfy both criteria:
29, 65, 101, ...
29/6 = 4 remainder 5
65/6 = 10 remainder 5
101/6 = 16 remainder 5
We see that the remainder is always 5. The two statements together are sufficient to answer the question.
Answer: C
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