91Ó°ÊÓ

The purpose of this exploration is to investigate the possibilities for which integers cannot be the sum of the cubes of two or three integers. (a) If \(x\) is an integer, what are the possible values (between 0 and 8 , inclusive) for \(x^{3}\) modulo \(9 ?\) (b) If \(x\) and \(y\) are integers, what are the possible values for \(x^{3}+y^{3}\) (between 0 and 8 , inclusive) modulo \(9 ?\) (c) If \(k\) is an integer and \(k \equiv 3(\bmod 9), \operatorname{can} k\) be equal to the sum of the cubes of two integers? Explain. (d) If \(k\) is an integer and \(k \equiv 4(\bmod 9), \operatorname{can} k\) be equal to the sum of the cubes of two integers? Explain. (e) State and prove a theorem of the following form: For each integer \(k\), if (conditions on \(k\) ), then \(k\) cannot be written as the sum of the cubes of two integers. Be as complete with the conditions on \(k\) as possible based on the explorations in Part (b). (f) If \(x, y,\) and \(z\) are integers, what are the possible values (between 0 and 8 , inclusive) for \(x^{3}+y^{3}+z^{3}\) modulo \(9 ?\) (g) If \(k\) is an integer and \(k \equiv 4(\bmod 9),\) can \(k\) be equal to the sum of the cubes of three integers? Explain. (h) State and prove a theorem of the following form: For each integer \(k\), if (conditions on \(k\) ), then \(k\) cannot be written as the sum of the cubes of three integers. Be as complete with the conditions on \(k\) as possible based on the explorations in Part (f).

Is the following proposition true or false? Justify your conclusion with a proof or a counterexample. For each natural number \(n\), if 3 does not divide \(\left(n^{2}+2\right),\) then \(n\) is not a prime number or \(n=3\).

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true. (a) For all real numbers \(x\) and \(y, \sqrt{x y} \leq \frac{x+y}{2}\). (b) For all real numbers \(x\) and \(y, x y \leq\left(\frac{x+y}{2}\right)^{2}\). (c) For all nonnegative real numbers \(x\) and \(y, \sqrt{x y} \leq \frac{x+y}{2}\).

Consider the following proposition: There are no integers \(a\) and \(b\) such that \(b^{2}=4 a+2\) (a) Rewrite this statement in an equivalent form using a universal quantifier by completing the following: For all integers \(a\) and \(b, \ldots\) (b) Prove the statement in Part (a).

Is the following statement true or false? Justify your conclusion. For each integer \(n\) that is greater than 1 , if \(a\) is the smallest positive factor of \(n\) that is greater than \(1,\) then \(a\) is prime. See Exercise (13) in Section 2.4 (page 78 ) for the definition of a prime number and the definition of a composite number.

Is the following proposition true or false? Justify your conclusion with a counterexample or a proof. For each integer \(a, 3\) divides \(a^{3}+23 a\).

Using only the digits 1 through 9 one time each, is it possible to construct a 3 by 3 magic square with the digit 3 in the center square? That is, is it possible to construct a magic square of the form $$ \begin{array}{|c|c|c|} \hline a & b & c \\ \hline d & 3 & e \\ \hline f & g & h \\ \hline \end{array} $$ where \(a, b, c, d, e, f, g, h\) are all distinct digits, none of which is equal to 3 ? Either construct such a magic square or prove that it is not possible.

Pythagorean Triples. Three natural numbers \(a, b,\) and \(c\) with \(a

Consider the following proposition: Proposition. For all integers \(m\) and \(n,\) if \(n\) is odd, then the equation $$ x^{2}+2 m x+2 n=0 $$ has no integer solution for \(x\). (a) What are the solutions of the equation when \(m=1\) and \(n=-1 ?\) That is, what are the solutions of the equation \(x^{2}+2 x-2=0 ?\) (b) What are the solutions of the equation when \(m=2\) and \(n=3\) ? That is, what are the solutions of the equation \(x^{2}+4 x+6=0 ?\) (c) Solve the resulting quadratic equation for at least two more examples using values of \(m\) and \(n\) that satisfy the hypothesis of the proposition. (d) For this proposition, why does it seem reasonable to try a proof by contradiction? (e) For this proposition, state clearly the assumptions that need to be made at the beginning of a proof by contradiction. (f) Use a proof by contradiction to prove this proposition.

The Last Two Digits of a Large Integer. Notice that \(7,381,272 \equiv 72(\) mod 100\()\) since \(7,381,272-72=7,381,200\). which is divisible by 100 . In general, if we start with an integer whose decimal representation has more than two digits and subtract the integer formed by the last two digits, the result will be an integer whose last two digits are \(00 .\) This result will be divisible by 100 . Hence, any integer with more than 2 digits is congruent modulo 100 to the integer formed by its last two digits. (a) Start by squaring both sides of the congruence \(3^{4} \equiv 81\) (mod 100 ) to prove that \(3^{8}=61\) (mod 100 ) and then prove that \(3^{16}=21\) (mod 100 ). What does this tell you about the last two digits in the decimal representation of \(3^{16} ?\) (b) Use the two congruences in Part (24a) and laws of exponents to determine \(r\) where \(3^{20} \equiv r(\bmod 100)\) and \(r \in \mathbb{Z}\) with \(0 \leq r<100 .\) What does this tell you about the last two digits in the decimal representation of \(3^{20} ?\) (c) Determine the last two digits in the decimal representation of \(3^{400}\). (d) Determine the last two digits in the decimal representation of \(4^{804}\). Hint: One way is to determine the "mod 100 values" for \(4^{2}, 4^{4}, 4^{8}\), \(4^{16} \cdot 4^{32}, 4^{64},\) and so on. Then use these values and laws of exponents to determine \(r,\) where \(4^{804} \equiv r(\) mod 100\()\) and \(r \in \mathbb{Z}\) with \(0 \leq r<\) \(100 .\) (e) Determine the last two digits in the decimal representation of \(3^{3356}\). (f) Determine the last two digits in the decimal representation of \(7^{403}\).