91Ó°ÊÓ

a. Use the quotient-remainder theorem with \(d=3\) to prove that the product of any three consecutive integers is divisible by \(3 .\) b. Use the mod notation to rewrite the result of part (a).

. If \(n\) is any even integer, then \((-1)^{2}=1\).

Prove each of the statements in 23-29. For any odd integer \(n\), $$ \left[\frac{n^{2}}{4}\right]=\frac{n^{2}+3}{4} $$

Prove that if one solution for a quadratic equation of the form \(x^{2}+b x+c=0\) is rational (where \(b\) and \(c\) are ra-tional), then the other solution is also rational. (Use the fact that if the solutions of the equation are \(r\) and \(s\), then \(\left.x^{2}+b x+c=(x-r)(x-s) .\right)\)

A fast-food chain has a contest in which a card with numbers on it is given to each customer who makes a purchase. If some of the numbers on the card add up to 100 , then the customer wins \(\$ 100\). A certain customer receives a card containing the numbers $$ 72,21,15,36,69,81,9,27,42, \text { and } 63 . $$ Will the customer win \(\$ 100\) ? Why or why not?

Definition: The least common multiple of two nonzero integers \(a\) and \(b\), denoted \(\operatorname{lcm}(a, b)\), is the positive integer \(c\) such that a. \(a \mid c\) and \(b \mid c\) b. for all integers \(m\), if \(a \mid m\) and \(b \mid m\), then \(c \mid m\). Prove that for all positive integers \(a\) and \(b\). \(\operatorname{gcd}(a, b) \cdot \operatorname{lcm}(a, b)=a b\).

Prove that for all integers \(n\), if \(n>2\) then there is a prime number \(p\) such that \(n

Prove that if a real number \(c\) satisfies a polynomial equation of the form $$ r_{3} x^{3}+r_{2} x^{2}+r_{1} x+r_{0}=0, $$ where \(r_{0}, r_{1}, r_{2}\), and \(r_{3}\) are rational numbers, then \(c\) satisfies an equation of the form $$ n_{3} x^{3}+n_{2} x^{2}+n_{1} x+n_{0}=0 \text {, } $$ where \(n_{0}, n_{1}, n_{2}\), and \(n_{3}\) are integers.

Is it possible to have a combination of nickels, dimes, and quarters that add up to \(\$ 4.72 ?\) Explain.

a. Fermat's last theorem says that for all integers \(n>2\), the equation \(x^{n}+y^{n}=z^{n}\) has no positive integer solution (solution for which \(x, y\), and \(z\) are positive integers). Prove the following: If for all prime numbers \(p>2\), \(x^{p}+y^{p}=z^{p}\) has no positive integer solution, then for any integer \(n>2\) that is not a power of \(2, x^{n}+y^{n}=z^{n}\) has no positive integer solution. b. Fermat proved that there are no integers \(x, y\), and \(z\) such that \(x^{4}+y^{4}=z^{4}\). Use this result to remove the restriction in part (a) that \(n\) not be a power of 2 . That is, prove that if \(n\) is a power of 2 and \(n>4\), then \(x^{n}+y^{n}=z^{n}\) has no positive integer solution.