Problem Solving

kakz wrote:The difference between positive two-digit integer A and the smaller two-digit integer B is twice A's units digit. What is the hundreds digit of the product of A and B?

(1) The tens digit of A is prime.
(2) Ten is not divisible by the tens digit of A.[/spoiler]

There is no need to check for so many possibilities.

Say, A = (10a + b), where a is the tens digit and b is the units digit of a.
Hence, B must be (10a - b)

Thus, AB = (10a + b)(10a - b) = (100a² - b²)

Note that, 1 ≤ b ≤ 9
So, 1 ≤ b² ≤ 81

Statement 1: Possible values of a : 2, 3, 5, and 7
Say, a = 2 and b = 1 --> AB = (100*4 - 1) = 399 --> Hundreds digit is 3
Say, a = 3 and b = 1 --> AB = (100*9 - 1) = 899 --> Hundreds digit is 8

NOT Sufficient

Statement 2: Possible values of a : 1, 3, 4 etc
Say, a = 1 and b = 1 --> AB = (100*1 - 1) = 099 --> Hundreds digit is 0
Say, a = 3 and b = 1 --> AB = (100*9 - 1) = 899 --> Hundreds digit is 8

NOT Sufficient

1 & 2 Together: Possible values of a : 3 and 7
Say, a = 3 and 1 ≤ b² ≤ 81 --> (900 - 81) ≤ AB ≤ (900 - 1) --> 819 ≤ b² ≤ 899 --> Hundreds digit is always 8
Say, a = 7 and 1 ≤ b² ≤ 81 --> (4900 - 81) ≤ AB ≤ (4900 - 1) --> 4819 ≤ b² ≤ 4899 --> Hundreds digit is always 8

SUFFICIENT

The correct answer is C.

kakz wrote:The difference between positive two-digit integer A and the smaller two-digit integer B is twice A's units digit. What is the hundreds digit of the product of A and B?

(1) The tens digit of A is prime.
(2) Ten is not divisible by the tens digit of A.

OA is C

Target question: What is the hundreds digit of the product AB?

Given: The difference between positive two-digit integer A and the smaller two-digit integer B is twice A's units digit
Let x = the tens digit of A, and let y = the units digit of A
So, the VALUE of A = 10x + y

From the given information, we can write: (10x + y) - B = 2y
Add B to both sides: 10x + y = 2y + B
Subtract 2y from both sides: 10x - y = B

So, A = 10x + y and B = 10x - y
So, the product AB = (10x + y)(10x - y) = 100x² - y²


Statement 1: The tens digit of A is prime.
In other words, x is prime
Let's TEST some values.
Case a: x = 2 (which is prime) and y = 3. In this case, AB = 100(2²) - 3² = 400 - 9 = 391. So, the answer to the target question is the hundreds digit of AB is 3
Case b: x = 3 (which is prime) and y = 1. In this case, AB = 100(3²) - 1² = 900 - 1 = 899. So, the answer to the target question is the hundreds digit of AB is 8
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Ten is not divisible by the tens digit of A.
10 is not divisible by 3, 4, 6, 7, 8, or 9
In other words, x could equal 3, 4, 6, 7, 8, or 9
Let's TEST some values.
Case a: x = 3 and y = 1. In this case, AB = 100(3²) - 1² = 900 - 1 = 899. So, the answer to the target question is the hundreds digit of AB is 8
Case b: x = 6 and y = 1. In this case, AB = 100(6²) - 1² = 3600 - 1 = 3599. So, the answer to the target question is the hundreds digit of AB is 5
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x could equal 2, 3, 5 or 7
Statement 2 tells us that x could equal 3, 4, 6, 7, 8, or 9
When we COMBINE the two statements, we see that x must equal EITHER 3 OR 7

IMPORTANT: Many students will incorrectly conclude that, since x can equal EITHER 3 OR 7, then the combined statements are not sufficient.
However, the target question is not asking us for the value of x; the target question is asking for the hundreds digit of AB.
So, let's test the two possible values of x:
Case a: x = 3 and y = any single digit. In this case, AB = 100(3²) - (any single digit)² = 900 - (some number less than 100) = 8??. So, the answer to the target question is the hundreds digit of AB is 8
Case b: x = 7 and y = any single digit. In this case, AB = 100(7²) - (any single digit)² = 4900 - (some number less than 100) = 48??. So, the answer to the target question is the hundreds digit of AB is 8
Aha!!!
In both possible cases, the answer to the target question is the SAME.
So, it MUST be the case that the hundreds digit of AB is 8
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent