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.
