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Prove each of the following propositions:

If $n$ is an integer greater than 2, then for all integers $m$, $n$ does not divide $m$ or $n + m \ne nm$.

Prove each of the following propositions:

For all real numbers $a$ and $b$, if $a > 0$ and $b > 0$, then

$$\frac{2}{a} + \frac{2}{b} \ne \frac{4}{a + b}$$

.

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.

Hint: Three consecutive natural numbers can be represented by $n$, $n + 1$, and $n + 2$, where $n \in \mathbb{N}$, or three consecutive natural numbers can be represented by $m - 1$, $m$, and $m + 1$, where $m \in \mathbb{N}$.

Three natural numbers $a$, $b$, and $c$ with $a \lt b \lt c$ are called a Pythagorean triple

provided that $a^2 + b^2 = c^2$. For example, the numbers 3, 4, and 5 form a Pythagorean triple, and the numbers 5, 12, and 13 form a Pythagorean triple.

Verify that if $a = 20$, $b = 21$, and $c = 29$, then $a^2 + b^2 = c^2$, and hence, 20, 21, and 29 form a Pythagorean triple.

Three natural numbers $a$, $b$, and $c$ with $a \lt b \lt c$ are called a Pythagorean triple

provided that $a^2 + b^2 = c^2$. For example, the numbers 3, 4, and 5 form a Pythagorean triple, and the numbers 5, 12, and 13 form a Pythagorean triple.

Determine two other Pythagorean triples. That is, find integers $a, b$, and $c$ such that $a^2 + b^2 = c^2$.

Three natural numbers $a$, $b$, and $c$ with $a \lt b \lt c$ are called a Pythagorean triple

provided that $a^2 + b^2 = c^2$. For example, the numbers 3, 4, and 5 form a Pythagorean triple, and the numbers 5, 12, and 13 form a Pythagorean triple.

Is the following proposition true or false? Justify your conclusion. For all integers $a$, $b$, and $c$, if $a^2 + b^2 = c^2$, then $a$ is even or $b$ is even.

Consider the following proposition: There are no integers $a$ and $b$ such that $b^2 = 4a + 2$.

Rewrite this statement in an equivalent form using a universal quantifier by completing the following:

For all integers

a

and

b

,

.

A magic square

is a square array of natural numbers whose rows, columns, and diagonals all sum to the same number. For example, the following is a 3 by 3 magic square since the sum of the 3 numbers in each row is equal to 15, the sum of the 3 numbers in each column is equal to 15, and the sum of the 3 numbers in each diagonal is equal to 15.

| 8 | 3 | 4 | | --- | --- | --- | | 1 | 5 | 9 | | 6 | 7 | 2 |

Prove that the following 4 by 4 square cannot be completed to form a magic square.

| | 1 | | 2 | | --- | --- | --- | --- | | 3 | 4 | 5 | | | 6 | 7 | | 8 | | 9 | | 10 | |

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

| $a$ | $b$ | $c$ | | --- | --- | --- | | $d$ | 3 | $e$ | | $f$ | $g$ | $h$ |

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.