Question

What are the values of three integers \( a, b, \) and \( c \)? I. \( ab = 8 \)
I II. \( bc = 9 \)

• A. If the question can be answered with the help of statement I alone
• B. If the question can be answered with the help of statement II alone
• C. If both, statement I and statement II are needed to answer the question
• D. If the question cannot be answered even with the help of both the statements

Hint:

If you have more variables than equations, you often need all statements to narrow down the solution.

Answer 1

The Correct Option is C

We are given two equations involving three variables:
I. \( ab = 8 \)
II. \( bc = 9 \) From I: Possible integer factor pairs for 8 are:
\( (1, 8), (2, 4), (-1, -8), (-2, -4), (4, 2), (8, 1), (-4, -2), (-8, -1) \) From II: Factor pairs for 9 are:
\( (1, 9), (3, 3), (-1, -9), (-3, -3), (9, 1), (-9, -1) \) To determine values of \( a, b, c \), we need a common value of \( b \) from both statements. Example: Try \( b = 2 \): from I, \( a = 4 \) (since \( ab = 8 \)); from II, \( c = 4.5 \) (non-integer) Try \( b = 3 \): then from I, \( a = 8/3 \) (not integer) Try \( b = 1 \): then \( a = 8 \), \( c = 9 \) → all integers, works!
So one possible solution: \( a = 8, b = 1, c = 9 \) Try other consistent values to ensure uniqueness.
Check \( b = -1 \): then \( a = -8 \), \( c = -9 \) → valid integer triple.
So multiple valid integer triplets exist. But unless both statements are used together, no unique solution for all three values can be determined.
Therefore, we need both statements together to identify valid sets of integer values for \( a, b, c \).