Data Sufficiency

BTGmoderatorLU wrote:Source: Manhattan GMAT

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.

The OA is C.

Say A is xy; thus, A = 10x + y

Since B is smaller than A by twice A's units digit, we have B = A - 2y = 10x + y - 2y = 10x -y

So, we have

A = 10x +y
B = 10x - y

We have to get the value of hundreds digit of the product of A and B = [(10x + y)*(10x - y) = 10x^2 - y^2]

Question: What is hundreds digit of 10x^2 - y^2?

Let's take each statement one by one.

(1) The tens digit of A is prime.

=> x: {2, 3, 5, 7}

Case 1: Say x = 2 (Minimum) and y = 1 (Minimum)
10x^2 - y^2 = 10*2^2 - 1^2 = 400 - 1 = 399. The hundreds digit is 3.

Case 2: Say x = 7 (Maximum) and y = 9 (Maximum)
10x^2 - y^2 = 10*7^2 - 9^2 = 4900 - 81 = 4819. The hundreds digit is 8.

No unique answer. Insufficient.

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

=> 10 is not divisible by x

=> x: {3, 4, 6, 7, 8, 9}

Case 3: Say x = 4 and y = 1
10x^2 - y^2 = 10*4^2 - 1^2 = 1600 - 1 = 1599. The hundreds digit is 5.

We already know from Case 2 of Statement 1 that at x = 7, the hundreds digit is 8 ≠ 5.

No unique answer. Insufficient.

(1) and (2) together

=> x: { 3, 7}

Case 4: Say x = 3 and y = 1 (Minimum)
10x^2 - y^2 = 10*3^2 - 1^2 = 900 - 1 = 899. The hundreds digit is 8.

We already know from Case 3 of Statement 1 that at x = 7, the hundreds digit is 8.

The value of the units digit y has no role to determine the hundreds digit. You may observe in that in each of the four cases, the units digit of 10x^2 - y^2 is 9. Additionally, you may try with 1) x = 3 and y = 9 (Maximum) and 2) x = 7 and y = 9 (Maximum). You will find that in each case the hundreds digit is 8. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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