Question

Each of the following questions is followed by two statements, I and II. Answer as follows:
Mark option (1) if the question can be answered using statement I alone, but not using statement II alone.
Mark option (2) if the question can be answered using statement II alone, but not using statement I alone.
Mark option (3) if the question can be answered using both statements I and II together, but not by using either statement alone.
Mark option (4) if the question cannot be answered even using both statements I and II together.

If a and b are integers, is \( \left(\dfrac{a}{4} + \dfrac{b}{5}\right) \) an integer?
I. The cube root of a is an even number which is 1/10th the value of b.
II. a is divisible by 5 and b is divisible by 4.

• A. The question can be answered using statement I alone, but not using statement II alone.
• B. The question can be answered using statement II alone, but not using statement I alone.
• C. The question can be answered using both statements I and II together, but not by using either statement alone.
• D. The question cannot be answered even using both statements I and II together.

Hint:

Turn the cube root condition into a = 8k^3, b = 20k for integer k, then check if a/4 + b/5 stays an integer for every k; for statement II, try two different numbers that satisfy it and see if you get different answers.

Answer 1

The Correct Option is A

Step 1: Restate what we need to find.
We are told \(a\) and \(b\) are integers, and we must decide whether \( \frac{a}{4} + \frac{b}{5} \) is always an integer, always not an integer, or it depends, under each statement.

Step 2: Translate statement I into equations.
Statement I says the cube root of \(a\) is an even number equal to one tenth of \(b\).
Let the cube root of \(a\) be \(c\), so \( a = c^3 \), and \(c\) is even.
Saying this cube root is one tenth the value of b means \( c = \frac{b}{10} \), so \( b = 10c \).
Since \(c\) is even, write \( c = 2k \) for some integer \(k\).
Then \( a = c^3 = (2k)^3 = 8k^3 \) and \( b = 10c = 10(2k) = 20k \).

Step 3: Substitute into the expression.
\[ \frac{a}{4} + \frac{b}{5} = \frac{8k^3}{4} + \frac{20k}{5} = 2k^3 + 4k \]
Since \(k\) is an integer, both \(2k^3\) and \(4k\) are integers, and the sum of two integers is always an integer.
So no matter what integer value \(k\) takes, \( \frac{a}{4} + \frac{b}{5} \) always comes out an integer. Statement I alone answers the question with a definite yes.

Step 4: Check statement II alone.
Statement II says \(a\) is divisible by 5 and \(b\) is divisible by 4. This does not say anything about \(a\) being divisible by 4 or \(b\) being divisible by 5.
Try \( a = 5, b = 4 \): both conditions hold, but \( \frac{a}{4} + \frac{b}{5} = 1.25 + 0.8 = 2.05 \), not an integer.
Try \( a = 20, b = 20 \): both conditions hold, and \( \frac{a}{4} + \frac{b}{5} = 5 + 4 = 9 \), an integer.
Since the same statement allows both an integer and a non-integer result, statement II alone cannot settle the question.

Final Answer:
Statement I alone is sufficient; statement II alone is not.
\[ \boxed{\text{Option (1)}} \]