If W Y X Z are the digits of the 4 digit number N,a positive integer, what is the remainder when N is divided by 9?
1. W+Y+Z+X= 13
2. N+5 is divisible by 9
D
DS Remainders
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- neelgandham
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If W Y X Z are the digits of the 4 digit number N,a positive integer, what is the remainder when N is divided by 9?
N = 1000W + 100Y + 10Z + X
N = 999W + 99Y + 9Z + W + Y + Z + X
N = 9*K + W + Y + Z + X
So, the remainder of N/9 is same as the remainder of W + Y + Z + X/9.
The remainder of 4! So, statement I is sufficient to answer the question.
N + 5 = 9K
N = 9K - 5
N = 9K - 9 + 4
N = 9*Integer + 4
The remainder of 4! So, statement I is sufficient to answer the question.
Answer : D
1. W+Y+Z+X= 13
N = 1000W + 100Y + 10Z + X
N = 999W + 99Y + 9Z + W + Y + Z + X
N = 9*K + W + Y + Z + X
So, the remainder of N/9 is same as the remainder of W + Y + Z + X/9.
The remainder of 4! So, statement I is sufficient to answer the question.
N + 5 is divisible by 92. N+5 is divisible by 9
N + 5 = 9K
N = 9K - 5
N = 9K - 9 + 4
N = 9*Integer + 4
The remainder of 4! So, statement I is sufficient to answer the question.
Answer : D
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Statement 1: W+Y+Z+X= 13avada wrote:If W Y X Z are the digits of the 4 digit number N,a positive integer, what is the remainder when N is divided by 9?
1. W+Y+Z+X= 13
2. N+5 is divisible by 9
D
If the sum of an integer's digits is a multiple of 9, then the INTEGER ITSELF is a multiple of 9.
Here, the sum of the digits is 13 -- 4 MORE than a multiple of 9.
The implication is that integer N = (multiple of 9) + 4.
(For a proof, see below.)
Thus, when N is divided by 9, the REMAINDER is 4.
SUFFICIENT.
Statement 2: N+5 is divisible by 9
N+5 = 9, 18, 27, 36...
Subtracting 5 from each value in this list, we get:
N = 4, 13, 22, 31...
In each case, when N is divided by 9, the remainder is 4:
4/9 = 0 R4.
13/9 = 1 R4.
22/9 = 2 R4.
SUFFICIENT.
The correct answer is D.
Proof for statement 1:
N = 1000W + 100Y + 10X + Z
= 999W + 99Y + 9X + (W+Y+X+Z)
= 9(111W + 11Y + X) + 13
= (multiple of 9) + (9 + 4)
= (multiple of 9 + 9) + 4
= (multiple of 9) + 4.
Thus, when N is divided by 9, the remainder is 4.
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So for a takeaway we can say in DS questions if they ask about remainders and they give us the sum like this we should know it's sufficient?GMATGuruNY wrote:If the sum of an integer's digits is a multiple of 9, then the INTEGER ITSELF is a multiple of 9.avada wrote:If W Y X Z are the digits of the 4 digit number N,a positive integer, what is the remainder when N is divided by 9?
1. W+Y+Z+X= 13
2. N+5 is divisible by 9
D
Here, the sum of the digits is 13 -- 4 MORE than a multiple of 9.
The implication is that integer N = (multiple of 9) + 4.
(For a proof, see below.)
Thus, when N is divided by 9, the REMAINDER is 4.
EX:
If xyz is a positive three digit integer. what is the remainder of xyz when divided by 3?
1) x+y+z = 14. So, here we don't need to do any math, we just know the rule stated above.
Let me know if I'm on the right (or wrong) track.
Thanks
A useful website I found that has every quant OG video explanation:
https://www.beatthegmat.com/useful-websi ... tml#475231
https://www.beatthegmat.com/useful-websi ... tml#475231
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Yes, we can use the same reasoning for a multiple of 3.alex.gellatly wrote:So for a takeaway we can say in DS questions if they ask about remainders and they give us the sum like this we should know it's sufficient?GMATGuruNY wrote:If the sum of an integer's digits is a multiple of 9, then the INTEGER ITSELF is a multiple of 9.avada wrote:If W Y X Z are the digits of the 4 digit number N,a positive integer, what is the remainder when N is divided by 9?
1. W+Y+Z+X= 13
2. N+5 is divisible by 9
D
Here, the sum of the digits is 13 -- 4 MORE than a multiple of 9.
The implication is that integer N = (multiple of 9) + 4.
(For a proof, see below.)
Thus, when N is divided by 9, the REMAINDER is 4.
EX:
If xyz is a positive three digit integer. what is the remainder of xyz when divided by 3?
1) x+y+z = 14. So, here we don't need to do any math, we just know the rule stated above.
Let me know if I'm on the right (or wrong) track.
Thanks
If the sum of an integer's digits is a multiple of 3, then the integer itself is a multiple of 3.
Here, the sum of the digits is 14 -- 2 more than a multiple of 3.
Thus, when the integer is divided by 3, the remainder will be 2.
Proof:
100X + 10Y + Z
= 99X + 9Y + (X+Y+Z)
= 9(11X + Y) + 14
= (multiple of 3) + (12 + 2)
= (multiple of 3 + 12) + 2
= (multiple of 3 + multiple of 3) + 2
= (multiple of 3) + 2.
Thus, when the integer is divided by 3, the remainder will be 2.
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For more information, please email me (Mitch Hunt) at [email protected].
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As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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Does this only work for multiples of 3 and 9? Or can we use this same logic for other integers?GMATGuruNY wrote:Yes, we can use the same reasoning for a multiple of 3.alex.gellatly wrote:So for a takeaway we can say in DS questions if they ask about remainders and they give us the sum like this we should know it's sufficient?GMATGuruNY wrote:If the sum of an integer's digits is a multiple of 9, then the INTEGER ITSELF is a multiple of 9.avada wrote:If W Y X Z are the digits of the 4 digit number N,a positive integer, what is the remainder when N is divided by 9?
1. W+Y+Z+X= 13
2. N+5 is divisible by 9
D
Here, the sum of the digits is 13 -- 4 MORE than a multiple of 9.
The implication is that integer N = (multiple of 9) + 4.
(For a proof, see below.)
Thus, when N is divided by 9, the REMAINDER is 4.
EX:
If xyz is a positive three digit integer. what is the remainder of xyz when divided by 3?
1) x+y+z = 14. So, here we don't need to do any math, we just know the rule stated above.
Let me know if I'm on the right (or wrong) track.
Thanks
If the sum of an integer's digits is a multiple of 3, then the integer itself is a multiple of 3.
Here, the sum of the digits is 14 -- 2 more than a multiple of 3.
Thus, when the integer is divided by 3, the remainder will be 2.
Proof:
100X + 10Y + Z
= 99X + 9Y + (X+Y+Z)
= 9(11X + Y) + 14
= (multiple of 3) + (12 + 2)
= (multiple of 3 + 12) + 2
= (multiple of 3 + multiple of 3) + 2
= (multiple of 3) + 2.
Thus, when the integer is divided by 3, the remainder will be 2.
A useful website I found that has every quant OG video explanation:
https://www.beatthegmat.com/useful-websi ... tml#475231
https://www.beatthegmat.com/useful-websi ... tml#475231
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Alex have a look at this site. Focus on "The Divisibility Rules" and you will understand the topic better.alex.gellatly wrote:
Does this only work for multiples of 3 and 9? Or can we use this same logic for other integers?
https://www.mathsisfun.com/divisibility-rules.html