91Ó°ÊÓ

(a) Is the base 2 logarithm of \(32, \log _{2}(32),\) a rational number or an irrational number? Justify your conclusion. (b) Is the base 2 logarithm of \(3, \log _{2}(3),\) a rational number or an irrational number? Justify your conclusion.

(a) Use the result in Proposition 3.33 to help prove that the integer \(m=\) 5,344,580,232,468,953,153 is not a perfect square. Recall that an integer \(n\) is a perfect square provided that there exists an integer \(k\) such that \(n=k^{2} .\) Hint: Use a proof by contradiction. (b) Is the integer \(n=782,456,231,189,002,288,438\) a perfect square? Justify your conclusion.

Let \(n\) be a natural number. Prove each of the following: (a) For every integer \(a, a \equiv a(\bmod n)\). This is called the reflexive property of congruence modulo \(n\). (b) For all integers \(a\) and \(b,\) if \(a \equiv b(\bmod n),\) then \(b \equiv a(\bmod n)\). This is called the symmetric property of congruence modulo \(n\). (c) For all integers \(a, b,\) and \(c,\) if \(a \equiv b(\bmod n)\) and \(b \equiv c(\bmod n),\) then \(a \equiv c(\bmod n)\) This is called the transitive property of congruence modulo \(n\).

Prove that for each integer \(a\), if \(a^{2}-1\) is even, then 4 divides \(a^{2}-1\).

In Exercise (15) in Section 3.2, we proved that there exists a real number solution to the equation \(x^{3}-4 x^{2}=7\). Prove that there is no integer \(x\) such that \(x^{3}-4 x^{2}=7\)

Prove each of the following: (a) For each nonzero real number \(x,\left|x^{-1}\right|=\frac{1}{|x|}\). (b) For all real numbers \(x\) and \(y,|x-y| \geq|x|-|y|\) Hint: An idea that is often used by mathematicians is to add 0 to an expression "intelligently". In this case, we know that \((-y)+y=0\). Start by adding this "version" of 0 inside the absolute value sign of \(|x|\). (c) For all real numbers \(x\) and \(y, \| x|-| y|| \leq|x-y|\).

Prove the following proposition: If \(p, q \in \mathbb{Q}\) with \(p

(a) If an integer has a remainder of 6 when it is divided by 7 , is it possible to determine the remainder of the square of that integer when it is divided by \(7 ?\) If so, determine the remainder and prove that your answer is correct. (b) If an integer has a remainder of 11 when it is divided by 12 , is it possible to determine the remainder of the square of that integer when it is divided by \(12 ?\) If so, determine the remainder and prove that your answer is correct. (c) Let \(n\) be a natural number greater than 2. If an integer has a remainder of \(n-1\) when it is divided by \(n\), is it possible to determine the remainder of the square of that integer when it is divided by \(n ?\) If so, determine the remainder and prove that your answer is correct.

(a) Verify that the triangle inequality is true for several different real numbers \(x\) and \(y .\) Be sure to have some examples where the real numbers are negative. (b) Explain why the following proposition is true: For each real number \(r\), \(-|r| \leq r \leq|r|\) (c) Now let \(x\) and \(y\) be real numbers. Apply the result in Part (14b) to both \(x\) and \(y\). Then add the corresponding parts of the two inequalities to obtain another inequality. Use this to prove that \(|x+y| \leq|x|+|y|\)

Prove that there do not exist three consecutive natural numbers such that the cube of the largest is equal to the sum of the cubes of the other two.