Permutations & Combinations
Grade 11
Question:
<p>The number of representations of the number 7056 as a product of 2 factors, is</p>
<p style="display:inline">8</p>
<p style="display:inline">45</p>
<p style="display:inline">23</p>
<p style="display:inline">18</p>
Step-by-Step Solution
Key Concept: To find the number of ways to express a perfect square as a product of two factors, calculate the total number of divisors $n$ and use the formula $(n+1)/2$ to include the square root case.
<p>7056 = 2<sup>4</sup> <span class="math-tex">$\times$</span> 7<sup>2</sup> <span class="math-tex">$\times$</span> 3<sup>2</sup><br />
7056 is to be represented as a product of 2 factors. These two factors are nothing but two divisors of 7056.<br />
The number of divisors = (4 + 1) (2 + 1) (2 + 1) = 45<br />
Out of 45 divisors, 44 divisors can be paired giving 22 representations<br />
45<sup>th</sup> divisor is 2<sup>2</sup> <span class="math-tex">$\times$</span> 7 <span class="math-tex">$\times$</span> 3 and 7056 = (2<sup>2</sup> <span class="math-tex">$\times$</span> 7 <span class="math-tex">$\times$</span> 3) (2<sup>2</sup> <span class="math-tex">$\times$</span> 7 <span class="math-tex">$\times$</span> 3)<br />
So, in all there are 23 representations.</p>
Correct Answer: C