CAT #3, Question 5

[jroach31]

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).

[RonPurewal]

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Thanks.