Real numbers
Grade Class 10
Question:
<p>If p<sub>1</sub> and p<sub>2</sub> are two odd prime numbers such that <span class="math-tex">\(p_1 > p_2\)</span>, then <span class="math-tex">\(p_{1}^{2}-p_{2}^{2}\)</span> is</p>
<p style="display:inline">an odd prime number</p>
<p style="display:inline">a prime number</p>
<p style="display:inline">an odd number</p>
<p style="display:inline">an even number</p>
Step-by-Step Solution
Key Concept: Apply the difference of squares identity and use the rule that adding or subtracting two odd numbers results in even numbers.
<p>Let p<sub>1</sub> and p<sub>2</sub> be 5 two odd primes.<br />
Then,<br />
<span class="math-tex">\(p_{1}^{2}-p_2^2=(p_1-p_2)(p_1+p_2)\)</span> <br />
We know that sum and difference of two odd numbers is even<br />
<span class="math-tex">\(\therefore\)</span> <span class="math-tex">\((p_1-p_2)~ and~(p_1+p_2)\)</span> are even numbers.<br />
Also, we know that product of even numbers is an even number, therefore<br />
<span class="math-tex">\(p_{1}^{2}-p_2^2=(p_1-p_2)(p_1+p_2)\)</span>, is an even number.</p>
Correct Answer: D