akhilsuhag wrote:Is the tens digit of a three-digit positive integer p divisible by 3?

(1) p-7 is a multiple of 3
(2) p-13 is a multiple of 3

Target question: Is the tens digit of a three-digit positive integer p divisible by 3?

IMPORTANT: When I scan the two statements, I see that they both tell me the SAME THING.
There's a rule that says: If N and K are both divisible by d, then N+K and N-K are also divisible by d.

OBSERVE: p-13 is 6 less than p-7. In other words, (p-7) - 6 = p-13

Statement 1 tells us that p-7 is divisible by 3, and we know that 6 is divisible by 3. So, by the above rule, p-13 must be divisible by 3.

When two statements provide the SAME information, we can conclude that the correct answer will be either D or E.

Statement 1: p-7 is a multiple of 3
There are several values of p that satisfy this condition. Here are two:
Case a: p = 139, in which case p-7 = 132, and 132 is divisible by 3. In this case, the tens digit of p (3) IS divisible by 3
Case b: p = 118, in which case p-7 = 111, and 111 is divisible by 3. In this case, the tens digit of p (1) is NOT divisible by 3
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: p-13 is a multiple of 3
We already learned that statements 1 and 2 provide the SAME information. So, if statement 1 is NOT SUFFICIENT, we know that statement 2 is NOT SUFFICIENT

If you're not convinced, check out these two conflicting cases:
Case a: p = 139, in which case p-13 = 126, and 126 is divisible by 3. In this case, the tens digit of p (3) IS divisible by 3
Case b: p = 118, in which case p-13 = 105, and 105 is divisible by 3. In this case, the tens digit of p (1) is NOT divisible by 3
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Since both statements are not sufficient, and since both statements provide the SAME information, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

Mathsbuddy wrote: Working backwards, firstly, for B to be a multiple of 3, it can only be 3, 6 or 9.

Hey Mathsbuddy,

You're absolutely right about the answer being E, but be careful. If B is a multiple of 3, then B can equal 0, 3, 6 or 9

Zero is a multiple of all integers. Likewise, zero is divisible by all integers.

Cheers,
Brent