<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-even-factors-with-pf">Even Factors with PF Exercise</a></p><p>There are two ways to do this.</p>
<p><span style="color:#e74c3c;"><span style="font-size:20px;">Way 1</span></span></p>
<p>Find the total number of positive factors and subtract the number of odd positive factors.</p>
<p style="text-align: center;"># of positive factors $-$ # of odd factors $=$ # of even factors</p>
<ul>
<li><strong>Example</strong>: How many even positive factors does $1$,$500$ have?
<ul>
<li>Prime Factorization: $=2^23^15^3$</li>
<li># of positive factors: $3 \times 2 \times 4 = 24$</li>
<li># of positive odd factors: $2 \times 4 = 8$</li>
<li># of positive even factors: $24 -8=16$</li>
</ul>
</li>
</ul>
<p><span style="color:#e74c3c;"><span style="font-size:20px;">Way 2</span></span></p>
<p>First prime factorize the number in question. Add $1$ to the exponent of every odd prime divisor, but <strong><u>do nothing</u></strong> to the exponent on the $2$. Multiply those numbers together.</p>
<ul>
<li><strong>Example</strong>: How many even positive factors does $1$,$500$ have?
<ul>
<li>Prime Factorization: $=2^23^15^3$</li>
<li>add $1$ to exponent of each odd prime divisor: $(1+1), (3+1) = 2, 4$</li>
<li>Do nothing to the exponent on the $2$: it remains $2$</li>
<li>Multiply those numbers together: $2 \times 2 \times 4 = 16$</li>
</ul>
</li>
</ul>
<p> </p>