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Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
When positive integer N is divided by 18, the remainder is x
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05 Jun 2018, 07:54
05 Jun 2018, 07:54
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When positive integer N is divided by 18, the remainder is x.
When N is divided by 6, the remainder is y. Which of the following are possible values of x and y?
i) x = 9 and y = 3
ii) x = 16 and y = 2
iii) x = 13 and y = 7
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i,ii and iii
When N is divided by 6, the remainder is y. Which of the following are possible values of x and y?
i) x = 9 and y = 3
ii) x = 16 and y = 2
iii) x = 13 and y = 7
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i,ii and iii
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: When positive integer N is divided by 18, the remainder is x
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07 Jun 2018, 06:47
07 Jun 2018, 06:47
1
GreenlightTestPrep wrote:
When positive integer N is divided by 18, the remainder is x. When N is divided by 6, the remainder is y. Which of the following are possible values of x and y?
i) x = 9 and y = 3
ii) x = 16 and y = 2
iii) x = 13 and y = 7
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i,ii and iii
i) x = 9 and y = 3
ii) x = 16 and y = 2
iii) x = 13 and y = 7
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i,ii and iii
Let's examine each statement separately...
i) x = 9 and y = 3
Let's come up with a value of N that satisfies this condition.
How about N = 9?
9 divided by 18 = 0 with remainder 9 (i.e., x = 9)
...and 9 divided by 6 = 1 with remainder 3 (i.e., y = 3)
Perfect, statement i is TRUE
Check the answer choices.....ELIMINATE D
ii) x = 16 and y = 2
Can you come up with a value of N that satisfies this condition?
How about N = 16?
16 divided by 18 = 0 with remainder 16 (i.e., x = 16, which WORKS)
However, 16 divided by 6 = 2 with remainder 4 (i.e., y = 4. NO GOOD)
How about N = 34?
34 divided by 18 = 1 with remainder 16 (i.e., x = 16, which WORKS)
However, 34 divided by 6 = 5 with remainder 4 (i.e., y = 4. NO GOOD)
We can keep testing N-values until we convince ourselves that there are no values of N that makes those values (x = 16 and y = 2) possible.
So, statement ii is FALSE
Check the answer choices.....ELIMINATE B and E
HOWEVER, if you need more convincing that statement ii is FALSE, we can make the following observations:
When positive integer N is divided by 18, the remainder is x: so, we can say that N = 18k + x for some integer k
When positive integer N is divided by 6, the remainder is y: so, we can say that N = 6j + y for some integer j
We can combine the two equations to get: 18k + x = 6j + y
Isolate x to get: x = y + 6j - 18k
Factor right side to get: x = y + 6(j - 3k)
Rewrite as: x = y + some multiple of 6
This means we'll never have the case where x = 16 and y = 2, because 16 CANNOT be written as 2 + some multiple of 6
So, statement ii is FALSE
iii) x = 13 and y = 7
We must be careful with this one.
While it is true that 13 CAN be written as 7 + some multiple of 6, we must also consider the following property of remainders:
When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0
Based on the above property, when we divide N by 6, the remainder can be 5, 4, 3, 2, 1, or 0
So, the remainder CANNOT be 7
In other words, y CANNOT equal 7
So, statement iii is FALSE
ELIMINATE C
Answer: A
RELATED VIDEO
Re: When positive integer N is divided by 18, the remainder is x
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15 Sep 2019, 12:51
15 Sep 2019, 12:51
1
I don't feel this approach is right, but I got the right answer. Brent, please let me know if this approach has any validity.
So, I said:
18Q + x = N = 6Q + y
18Q + x = 6Q + y
Then I plugged in numbers for x and y, for example:
18Q + 9 = 6Q + 3
Then I imagined Q was 1, so we get
27 = 9
It's plain that the two sides aren't equal, but they are multiples of one another.
I used the same thing to look at the other solutions, and saw they weren't multiples of one another.
I read your solution again, and I think I may have made a mistake in writing Q for both sides instead of Q sub 1 and Q sub 2. We are told N is the same, but we're not told Q is the same.
So, I guess I could say
18Q_1 + 9 = 6Q_2 + 3
Given Q_1 and Q_2 could be different values.
The other solutions won't be multiples of each other, and therefore can't be equal whatever Q_1 and Q_2 are. For example, if x = 13 and y = 17, we get
18Q_1 + 13 = 6Q_2 + 7
if both Qs are 1, we get
31 = 13, which are clearly not multiples of each other, and therefore no matter what the Q's are for both sides, the two sides will never be equal.
I found the same to be true for the other answer choices. None of them resulted in multiples.
Was on the right path? Or was getting the right answer just a coincidence?
PS Sorry, if my line of reasoning was confusing. It felt confusing to me writing it.
So, I said:
18Q + x = N = 6Q + y
18Q + x = 6Q + y
Then I plugged in numbers for x and y, for example:
18Q + 9 = 6Q + 3
Then I imagined Q was 1, so we get
27 = 9
It's plain that the two sides aren't equal, but they are multiples of one another.
I used the same thing to look at the other solutions, and saw they weren't multiples of one another.
I read your solution again, and I think I may have made a mistake in writing Q for both sides instead of Q sub 1 and Q sub 2. We are told N is the same, but we're not told Q is the same.
So, I guess I could say
18Q_1 + 9 = 6Q_2 + 3
Given Q_1 and Q_2 could be different values.
The other solutions won't be multiples of each other, and therefore can't be equal whatever Q_1 and Q_2 are. For example, if x = 13 and y = 17, we get
18Q_1 + 13 = 6Q_2 + 7
if both Qs are 1, we get
31 = 13, which are clearly not multiples of each other, and therefore no matter what the Q's are for both sides, the two sides will never be equal.
I found the same to be true for the other answer choices. None of them resulted in multiples.
Was on the right path? Or was getting the right answer just a coincidence?
PS Sorry, if my line of reasoning was confusing. It felt confusing to me writing it.
