91Ó°ÊÓ

The product of any four consecutive integers is divisible by 8 .

"Proof: Suppose \(r\) and \(s\) are rational numbers. If \(r+s\) is rational, then by definition of rational \(r+s=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\). Also since \(r\) and \(s\) are rational, \(r=i / j\) and \(s=m / n\) for some integers \(i, j, m\), and \(n\) with \(j \neq 0\) and \(n \neq 0\). It follows that \(r+s=i / j+m / n=\) \(a / b\), which is a quotient of two integers with a nonzero denominator. Hence it is a rational number. This is what was to be shown. \(^{.1}\)

The square of any integer has the form \(4 k\) or \(4 k+1\) for some integer \(k\).

How many zeros are at the end of \(45^{8} \cdot 88^{5}\) ? Explain how you can answer this question without actually computing the number. (Hint: \(10=2 \cdot 5 .\) )

Theorem: The sum of any two even integers equals \(4 k\) for some integer \(k\). "Proof: Suppose \(m\) and \(n\) are any two even integers. By definition of even, \(m=2 k\) for some integer \(k\) and \(n=2 k\) for some integer \(k\). By substitution, \(m+n=2 k+2 k=4 k\). This is what was to be shown."

In 39-56 determine whether the statement is true or false. Justify your answer with a proof or a counterexample, as appropriate. 39\. The product of any two odd integers is odd.

If \(n\) is an integer and \(n>1\), then \(n !\) is the product of \(n\) and every other positive integer that is less than \(n\). For example, \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\). a. Write \(6 !\) in standard factored form. b. Write \(20 !\) in standard factored form. c. Without computing the value of \((20 !)^{2}\) determine how many zeros are at the end of this number when it is written in decimal form. Justify your answer.

In a certain town \(2 / 3\) of the adult men are married to \(3 / 5\) of the adult women. Assume that all marriages are monogamous (no one is married to more than one other person). Also assume that there are at least 100 adult men in the town. What is the least possible number of adult men in the town? of adult women in the town?

The negative of any odd integer is odd.

The difference of any two odd integers is odd.