Question

If abc > 0, such that a,b,c are integers then which of the following must be true?

• A. \(\frac{a}{b}\lt0\)
• B. \(\frac{ab}{c}\gt0\)
• C. bc < 0
• D. a > bc

Answer 1

The Correct Option is B

The problem involves understanding the sign conditions of integers \(a\), \(b\), and \(c\) given \(abc > 0\). Let's analyze the options provided one by one.

Given: \(abc > 0\) 

This implies that the product of \(a\), \(b\), and \(c\) is positive. This can only happen if either all three integers are positive or if one is positive and the other two are negative.

1. \(\frac{a}{b} < 0\): This implies \(a\) and \(b\) must have opposite signs. However, if \(a\) and \(b\) have opposite signs, \(ab\) would be negative, contradicting the condition \(abc > 0\). Therefore, this is false.
2. \(\frac{ab}{c} > 0\): This implies \(\frac{ab}{c}\) is positive. Given \(abc > 0\), this is true. Since the product \(abc\) is positive, the product \(ab\) must have the same sign as \(c\), yielding a positive fraction \(\frac{ab}{c} > 0\).
3. bc < 0: This condition implies that \(b\) and \(c\) must have opposite signs. However, if \(bc\) is negative, \(abc\) wouldn't be positive as initially given. Therefore, this is false.
4. a > bc: This implies \(a\) is greater than the product of \(b\) and \(c\). However, this does not have to be true given \(abc > 0\). There isn't enough information to conclude that \(a\) is necessarily greater than \(bc\). It is not necessarily true.

Conclusion: The only option that must be true, based on the condition \(abc > 0\), is \(\frac{ab}{c} > 0\). Therefore, the correct option is:

\(\frac{ab}{c} > 0\)