The difference between positive two-digit integer A and the smaller two-digit integer B is twice A‘s units digit.

[BTGmoderatorDC]

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 C

Source: Manhattan Prep

[Jay@ManhattanReview]

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 C

Source: Manhattan Prep

Say A = 10x + y, where x = tens digit and y = units digit

Since we know that the difference between positive two-digit integer A and the smaller two-digit integer B is twice A‘s units digit, B = 10x + y – 2y = 10x – y

So, A = 10x + y, and B = 10x – y, where y ≠ 0, else A = B, which is not posible.

Thus, AB = (10x + y)(10x – y) = (10x)^2 – y^2 = 100x^2 – y^2

We know that AB is a three-digit no. and y > 0; thus, x > 1, else at x = 1, AB = 100x^2 – y^2 would be a 2-digit no. Also note that since 1 ≤ y ≤ 9, we have 1 ≤ y^2 ≤ 81.

Note that tens and the units digits of 100x^2 would be 00, and y^2 (min. 1 and max. 81) would be deducted; for that, we need to carry 1 from 100x^2; but in all cases, there is no role of the value of y to decide the hundreds digit of AB. Let's see how.

Case 1: Say x = 2 and y =1

Thus, AB = 100x^2 – y^2 = 100*2^2 – 1^2 = 400 – 1 = 399; hundreds digit = 3

Case 2: Say x = 2 and y =9

Thus, AB = 100x^2 – y^2 = 100*2^2 – 9^2 = 400 – 81 = 319; hundreds digit = Same 3. Irrespective of the value of y, the units digit of 100x^2 – y^2 would be same

Now let's see how the values of x affect the hundreds digit of AB.

Case 3: Say x = 3 and y = 5 (the value of y has no role to play)

Thus, AB = = 100x^2 – y^2 = 100*3^2 – 5^2 = 900 – 25 = 875; hundreds digit = 8

We see that the value of hundreds digit differ; in cases 1 and 2, it was 3.

So, the question is what's the units digit of x^2?

Let's take each statement one by one.

(1) The tens digit of A is prime.

=> x is one among 2, 3, 5, and 7. So, x^2 is one among 4, 9, 25, and 49. We see that units digit of x^2 can be 4, 5 or 9. No unique answer. Insufficient.

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

=> x is one among 3, 4, 6, 7, 8, and 9. So, x^2 is one among 9, 16, 36, 49, 64, and 81. We see that units digit of x^2 can be 1, 4, 6, or 9. No unique answer. Insufficient.

(1) and (2) together

From both statements, x is one among 3, and 7. So, x^2 is either 9 or 49. In either case, the units digit of x^2 is 9. -- a unique value. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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