Sets, Relations & Functions
Properties of Irrational Numbers
Grade 11
Question:
<p>If \(a\) is an irrational number which is divisible by \(b\), then the number \(b\)</p>
<p>(a) must be rational</p>
<p>(b) must be irrational</p>
<p>(c) may be rational or irrational</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: If a rational number divides an irrational number, it leads to a contradiction; therefore the divisor must be irrational.
<p>If $a$ is divisible by $b$, then $a = b \times k$ for some integer $k$.</p><p>Given $a$ is irrational. If $b$ were rational (and $b \neq 0$), then $a = b \times k$ would be the product of a rational number $b$ and an integer $k$ (which is rational), resulting in a rational number. This contradicts the fact that $a$ is irrational.</p><p>Therefore, $b$ must be irrational. The answer is (b).</p>
Correct Answer: B