Data Sufficiency: Integer Product and Sum

Question

We have two unknown positive integers $m$ and $n$ , whose product is less than 100.

There are two additional statement of facts available:

Statement 1: $mn$ is divisible by six consecutive integers $\{j, j + 1, \dots, j + 5\}$
Statement 2: $m + n$ is a perfect square.

Which of the two statements above, alone or in combination shall be sufficient to determine the numbers $m$ and $n$ ?

Accepted Answer

B. Statement 1: LCM of 6 consecutive integers $\{1, 2, 3, 4, 5, 6\}$ is 60. Since $mn < 100$, $mn = 60$. Multiple pairs exist $(1,60), (2,30), (3,20), (4,15), (5,12), (6,10)$. Insufficient alone. Statement 2: $m+n$ is a perfect square. Insufficient alone. Together: Among pairs with product 60, sum $6+10 = 16 = 4^2$. Only $(6,10)$ gives a perfect square sum. Sufficient together.

Answer

Each statement alone is sufficient to answer the question.

Answer

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Answer

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Answer

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.

Answer

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Data Sufficiency: Integer Product and Sum

Explanation

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