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number N, a positive integer

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number N, a positive integer

by sanju09 » Fri Oct 15, 2010 10:12 pm
If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

[1] w + x + y + z = 13.

[2] N + 5 is divisible by 9.


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by neerajkumar1_1 » Sat Oct 16, 2010 1:06 am
statement 1)
any combination of w x y z will always give a remainder of 4, since the sum is constant...
sufficient...

statement 2)

(N + 5)/9 = n/9 +5/9

n/9 will leave no remainder... 5/9 will always leave a remainder of 5
sufficient...

this was just for explanation... else its obvious since N + 5 is divisble by 9, so N will always leave a remainder of 5 when divided by 9...

Pick D
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by sanju09 » Sat Oct 16, 2010 2:23 am
neerajkumar1_1 wrote:statement 1)
any combination of w x y z will always give a remainder of 4, since the sum is constant...
sufficient...

statement 2)

(N + 5)/9 = n/9 +5/9

n/9 will leave no remainder... 5/9 will always leave a remainder of 5
sufficient...

this was just for explanation... else its obvious since N + 5 is divisble by 9, so N will always leave a remainder of 5 when divided by 9...

Pick D
If N + 5 is divisible by 9, why will N/9 leave no remainder?
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by clock60 » Sat Oct 16, 2010 2:48 am
i also think that the answer is D , but solve 2 st slightly different from above
(2) n+5 is divisible by 9
5 is not divisible by 9 and leave remainder 5. 5=9*0+5
so it means that n is also not divisible by 9, but n while division by 9 leaves remainder that if sum with 5 is divisible by 9
and the only remainder of n to be the sum divisible by 9 is 4
to prove it consider possible remaiders while division by 9. they are 1,2,3,4,5,6,7,8, and now sum each with 5
1+5=6 not divisible by 9
2+5=7 also not
3+5=8 not
4+5=9 yes so 4 out remaider
5+5=10 not
6+5=11 not
7+5=12 not
8+5=13 not
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by neerajkumar1_1 » Sat Oct 16, 2010 3:37 am
sanju09 wrote:
neerajkumar1_1 wrote:statement 1)
any combination of w x y z will always give a remainder of 4, since the sum is constant...
sufficient...

statement 2)

(N + 5)/9 = n/9 +5/9

n/9 will leave no remainder... 5/9 will always leave a remainder of 5
sufficient...

this was just for explanation... else its obvious since N + 5 is divisble by 9, so N will always leave a remainder of 5 when divided by 9...

Pick D
If N + 5 is divisible by 9, why will N/9 leave no remainder?
My bad,
wrote the answer in a hurry...
since (n+5)/9 is divisble by 9

it can be written as n/9 + 5/9

now since 5/9 will leave a remainder of 5
n/9 will have to leave a remainder of 4 such that the entire sum is divisible by 9

e.g n = 4 then 4+5 = 9
or n = 13 then 13+5= 18 so on so forth..



one can understand it another way...
since (n+5)/9 is divisible by 9
so n must be 5 short of multiple of 9
n=9-5 = 4
if n is 5 short of multiple of 9 then n will always leave a remainder of 4 when divided by 9...
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by Stuart@KaplanGMAT » Sat Oct 16, 2010 10:17 am
sanju09 wrote:If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

[1] w + x + y + z = 13.

[2] N + 5 is divisible by 9.


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A slightly different solution.

(1) 9 is a special number when it comes to remainders, similar to 3. To determine whether a number is divisible by 9, you simply add the digits. If the digits sum to 9, the number is a multiple of 9.

As a corollary to this rule, to find the remainder of any number divided by 9, you simply add the digits and subtract the closest (and smaller) multiple of 9.

For example, looking at the number 2483:

2+4+8+3 = 17, and 17-9 = 8; therefore 2483/9 has a remainder of 8.

(1) gives us the sum of the digits, so is sufficient.

(2) N + 5 is divisible by 9

When it comes to multiples, all numbers revolve in cycles. So, if N+5 is a multiple of 9:

(N+4)/9 has rem 8
(N+3)/9 has rem 7
(N+2)/9 has rem 6
(N+1)/9 has rem 5
N/9 has rem 4

Accordingly, (2) is also sufficient.

Note that we didn't actually need to figure out the answer - just recognizing that remainders work in cycles tells us that (2) is sufficient.

Each of (1) and (2) is sufficient alone: choose (D).

This question is a great example of how understanding number properties leads to quick and confident points on test day. If you apply the appropriate principles, this is a 10-15 second question; if you attack it by picking numbers, you can still get the question correct, but you spend considerably more time doing so.
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