Data Sufficiency - Number Properties

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
2. B. The Question can be answered by using either Statement alone
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
4. D. The Question cannot be answered even by using both the Statements together

Answer: D

Explanation

We need to determine if (x+y) is an integer.

**Statement I: (2x+y) is an integer.**
– Let x = 0.5, y = 1. Then 2x+y = 2(0.5)+1 = 1+1 = 2 (integer). But x+y = 0.5+1 = 1.5 (not an integer).
– Let x = 1, y = 1. Then 2x+y = 2(1)+1 = 3 (integer). And x+y = 1+1 = 2 (integer).
Since (x+y) can be an integer or not, Statement I alone is insufficient.

**Statement II: (x+2y) is an integer.**
– Let x = 1, y = 0.5. Then x+2y = 1+2(0.5) = 1+1 = 2 (integer). But x+y = 1+0.5 = 1.5 (not an integer).
– Let x = 1, y = 1. Then x+2y = 1+2(1) = 3 (integer). And x+y = 1+1 = 2 (integer).
Since (x+y) can be an integer or not, Statement II alone is insufficient.

**Combining Statement I and Statement II:**
We have: (2x+y) = K (an integer) and (x+2y) = M (an integer).
Adding these two equations: (2x+y) + (x+2y) = K + M
3x + 3y = K + M
3(x+y) = K + M.
Since K and M are integers, K+M is also an integer. So, 3(x+y) is an integer.
However, if 3(x+y) is an integer, (x+y) itself is not necessarily an integer. For example, if x=1/3 and y=1/3:
– 2x+y = 2/3 + 1/3 = 1 (integer).
– x+2y = 1/3 + 2/3 = 1 (integer).
– But x+y = 1/3 + 1/3 = 2/3 (not an integer).
Therefore, even with both statements combined, we cannot definitively say if (x+y) is an integer. This question tests data sufficiency and properties of integers and rational numbers.

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
2. B. The Question can be answered by using either Statement alone
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
4. D. The Question cannot be answered even by using both the Statements together

Answer: A

Explanation

Let the age of P be 10x + y and the age of Q be 10y + x, where x and y are digits from 1 to 9 (since ages are >10 and Q).**
10x + y > 10y + x => 9x > 9y => x > y.
Possible pairs (x,y) satisfying x>y and x,y are digits (y cannot be 0 as Q would be a single digit):
– If x=2, y=1 => P=21, Q=12. Difference = 9.
– If x=3, y=1 => P=31, Q=13. Difference = 18.
– If x=3, y=2 => P=32, Q=23. Difference = 9.
Since the difference is not unique, Statement I alone is insufficient.

**Statement II: The sum of their ages is 11/6 times their difference.**
P + Q = (11/6) * |P – Q|
(10x + y) + (10y + x) = (11/6) * |(10x + y) – (10y + x)|
11x + 11y = (11/6) * |9x – 9y|
11(x + y) = (11/6) * 9 * |x – y|
x + y = (9/6) * |x – y|
x + y = (3/2) * |x – y|
Since x and y are positive digits, x+y is positive. For the equation to hold, |x-y| must also be positive, implying x ≠ y. Also, for x+y to be a multiple of 3/2, x-y must be even. This means x and y must have the same parity. If x-y is positive, then x > y.
2(x + y) = 3(x – y)
2x + 2y = 3x – 3y
5y = x.
Since x and y are single digits (1-9) and y cannot be 0:
– If y=1, then x=5. This gives P = 51 and Q = 15. (P>Q is satisfied, 51>15).
– If y=2, then x=10 (not a single digit). No other solutions.
So, the only unique pair is (x=5, y=1), which means P=51 and Q=15.
The difference in their ages = P – Q = 51 – 15 = 36.
This uniquely determines the ages and their difference. Thus, Statement II alone is sufficient.

Therefore, the question can be answered by using Statement II alone, but not Statement I alone.

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
2. B. The Question can be answered by using either Statement alone.
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
4. D. The Question cannot be answered even by using both the Statements together.

Answer: D

Explanation

We need to find a unique 3-digit number.
Statement 1: Sum of digits = Product of digits. For example, if the digits are 1, 2, 3, then 1+2+3=6 and 1*2*3=6. Numbers like 123, 132, 213, 231, 312, 321 all satisfy this. Since there are multiple possibilities, Statement 1 alone is not sufficient.
Statement 2: The number is divisible by the sum of its digits. For example, 102 (sum=3, 102/3=34) and 108 (sum=9, 108/9=12) both satisfy this. Since there are multiple possibilities, Statement 2 alone is not sufficient.
Combining both statements: We look for numbers where the sum of digits equals the product of digits, AND the number is divisible by this sum. Using the example digits 1, 2, 3 (sum=6, product=6):
– 132: Sum=6, Product=6. 132 is divisible by 6 (132/6=22). This number satisfies both.
– 312: Sum=6, Product=6. 312 is divisible by 6 (312/6=52). This number also satisfies both.
Since both 132 and 312 satisfy both statements, we cannot determine a unique 3-digit number. Therefore, even using both statements together, the question cannot be answered.