这是一份leeds利兹大学MATH202701作业代写的成功案例
Let us make another calculation of a greatest common divisor using the prime factorization approach, as a guide to formulating precisely what the approach is. We will compute $(8316,19800)$. The prime factorizations are
$$
8316=2^{2} \times 3^{3} \times 7 \times 11
$$
and
$$
19800=2^{3} \times 3^{2} \times 5^{2} \times 11
$$
Let us rewrite these factorizations as
$$
8316=2^{2} \times 3^{3} \times 5^{0} \times 7^{1} \times 11^{1}
$$
and
$$
19800=2^{3} \times 3^{2} \times 5^{2} \times 7^{0} \times 11^{1} .
$$
MATH202701 COURSE NOTES ：
These three statements can be summarized in equations. For example, the first statement can be written as
$$
\mathcal{C}(0)+\mathcal{C}(1)=\mathcal{C}(1)
$$
Write the two other statements as equations of the same type.
Perform analogous steps for multiplication: What is the product of an even integer and an odd integer? An even integer and an even integer? An odd integer and an odd integer? Summarize your answers in three equations of the form
$$
\mathcal{C}(i) \times \mathcal{C}(j)=\mathcal{C}(k)
$$
Observe that your sum and product rules turn the collection
$$
{\mathcal{C}(0), \mathcal{C}(1)}
$$