Bunuel 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.
A student asked me to respond to this question, so here it goes...
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 3Case 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 8Since 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 8Case 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 5Since 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 8Case b: x = 7 and y = any single digit. In this case, AB = 100(7²) - (any single digit)² = 4900 - (some number less than 100) = 4
8??. So, the answer to the target question is
the hundreds digit of AB is 8Aha!!!
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 8Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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