Can someone please explain the explanation for evaluating statement #2? I don't understand the logic for why if X+7 is divisible by 9, X-2 is also divisible by 9. Thanks!
If a and b are the digits of the two-digit positive integer X, what is the remainder when X is divided by 9?
(1) a + b = 11
(2) X + 7 is divisible by 9
In order for a number to be divisible by 9, the sum of its digits must be divisible by 9.
(1) SUFFICIENT: The sum of the digits a and b here is not divisible by 9, so X is not divisible by 9. It turns out, however, that the sum of the digits can also be used to find the remainder. Since the sum of the digits here has a remainder of 2 when divided by 9, the number itself has a remainder of 2 when divided by 9.
We can use a few values for a and b to show that this is the case:
When a = 5 and b = 6, 56 divided by 9 has a remainder of 56 - 54 = 2
When a = 7 and b = 4, 74 divided by 9 has a remainder of 74 - 72 = 2
(2) SUFFICIENT: If X + 7 is divisible by 9, X - 2 would also be divisible by 9 (X - 2 + 9 = X + 7). If X - 2 is divisible by 9, then X itself has a remainder of 2 when divided by 9.
Again we could use numbers to prove this:
If X + 7 = 27, then X = 20, which has a remainder of 2 when divided by 9
If X + 7 = 18, then X = 11, which has a remainder of 2 when divided by 9
The correct answer is (D).
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Divisibility question
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Well as I understand the question it asks what is the remainder-
the second statement says-
(2) X + 7 is divisible by 9
Lets take x as 10a+b
we can put it as 10a+b+7 is divisible by 9
now if we plug in values for a and b (a is 1 and b is 1 gives us 10*1+1+7 is divisible by 9) therefore 10*1+1=11 thereby when divided by 9 we get 2 as remainder.
we can recheck by putting some other values which satisfy the second stmnt for the equation lets take 2 for a and 0 for b so when we add = 10*2+0+7=27 (again divisible by 9) and again our remainder is 2 if we divide 10*2+0 i.e. X by 9
Hence statement 2 is also sufficient.
the second statement says-
(2) X + 7 is divisible by 9
Lets take x as 10a+b
we can put it as 10a+b+7 is divisible by 9
now if we plug in values for a and b (a is 1 and b is 1 gives us 10*1+1+7 is divisible by 9) therefore 10*1+1=11 thereby when divided by 9 we get 2 as remainder.
we can recheck by putting some other values which satisfy the second stmnt for the equation lets take 2 for a and 0 for b so when we add = 10*2+0+7=27 (again divisible by 9) and again our remainder is 2 if we divide 10*2+0 i.e. X by 9
Hence statement 2 is also sufficient.
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Two rules about multiples:jroach31 wrote:Can someone please explain the explanation for evaluating statement #2? I don't understand the logic for why if X+7 is divisible by 9, X-2 is also divisible by 9. Thanks!
(multiple of y) + (multiple of y) = multiple of y.
(multiple of y) - (multiple of y) = multiple of y.
In the problem above:
x-2 = (x+7) - 9 = multiple of 9 - multiple of 9 = multiple of 9.
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Another way to see this:
If x + 7 is a multiple of 9, we can write an equation like x + 7 = 9 * something. (For convenience, let's call this something "k", so we'll have x + 7 = 9k.)
If x + 7 = 9k, then x = 9k - 7.
Since x = 9k - 7, we know that x - 2 = 9k - 7 - 2.
But then x - 2 = 9k - 9 = 9 * (k - 1). So x - 2 also equals 9 * something! (This time the "something" is k-1).
So x - 2 is ALSO a multiple of 9.
Intuitively this should make a lot of sense: since x + 7 and x - 2 are 9 units apart on the number line, if we one of them is a multiple of 9, the other must be too. (For instance, 18 is 9 units from 9 and 9 units from 27, both of which are multiples of 9.)
If x + 7 is a multiple of 9, we can write an equation like x + 7 = 9 * something. (For convenience, let's call this something "k", so we'll have x + 7 = 9k.)
If x + 7 = 9k, then x = 9k - 7.
Since x = 9k - 7, we know that x - 2 = 9k - 7 - 2.
But then x - 2 = 9k - 9 = 9 * (k - 1). So x - 2 also equals 9 * something! (This time the "something" is k-1).
So x - 2 is ALSO a multiple of 9.
Intuitively this should make a lot of sense: since x + 7 and x - 2 are 9 units apart on the number line, if we one of them is a multiple of 9, the other must be too. (For instance, 18 is 9 units from 9 and 9 units from 27, both of which are multiples of 9.)
