Exercises

Exercises 5.5 Exercises

1.

Let \(n\in\mathbb Z\text{.}\) Prove that if \(n^2+4n+5\) is odd, then \(n\) is even.

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2.

Let \(n\in\mathbb{Z}\text{.}\) Show that if \(5\nmid n^2\text{,}\) then \(5\nmid n\text{.}\)

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3.

Let \(n\in\mathbb{Z}\text{.}\) Prove that if \(5\nmid n\) or \(2\nmid n\text{,}\) then \(10\nmid n\text{.}\)

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4.

Let \(n,m \in \mathbb{N}\text{.}\) Prove that if \(n \neq 1\) and \(n \neq 2\text{,}\) then \(n\nmid m\) or \(n \nmid (m+2)\text{.}\)

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5.

Let \(n,m\in\mathbb{Z}\text{.}\) Prove that if \(n^2+m^2\) is even, then \(n,m\) have the same parity.

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6.

Let \(x\in\mathbb{R}\text{.}\) Show that if \(x^3+5x\geq x^2+1\text{,}\) then \(x \gt 0\text{.}\)

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7.

We say that the pair of numbers \(a,b\) are consecutive in the set \(S\) when \(a \lt b\) and there is no number \(c \in S\) so that \(a \lt c \lt b\text{.}\) That is, the number \(b\) is the next number in the set after \(a\text{.}\) For example:

\(5\) and \(6\) are consecutive integers.
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\(10\) and \(12\) are consecutive even numbers.
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\(25\) and \(30\) are consecutive multiples of \(5\text{.}\)
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Prove the following statement:

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Let \(a,b\in \mathbb{Z}\text{.}\) If \(a+b\) is not odd, then \(a\) and \(b\) are not consecutive.

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8.

Prove that if \(n\) is an even integer then \(n=4k\) or \(n=4k+2\) for some integer \(k\text{.}\)

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9.

Let \(n\in\mathbb Z\text{.}\) Show that \(2\mid (n^4-7)\) if and only if \(4\mid (n^2+3)\text{.}\)

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10.

Let \(a\in\mathbb{Z}\text{.}\) Prove that \(3\mid 5a\) if and only if \(3\mid a\text{.}\)

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11.

Let \(n\in\mathbb{Z}\text{.}\) Show that \((n^2-1)(n^2+2n)\) is divisible by \(4\text{.}\)

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12.

Prove the following statement:

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If \(x+y\) is odd, then either \(x\) or \(y\) is odd, but not both.

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13.

Let \(n\in\mathbb Z\text{.}\) Prove that if \(3\mid (n^2+4n+1)\text{,}\) then \(n\equiv 1\mod 3\text{.}\)

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14.

Let \(m\in\mathbb{Z}\text{.}\) Prove that if \(5\nmid m\text{,}\) then \(m^2\equiv 1 \mod{5}\) or \(m^2\equiv -1 \mod{5}\text{.}\)

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15.

Let \(q\in \mathbb{Z}\text{.}\) If \(3\nmid q\text{,}\) then \(q^2 \equiv 1 \mod 3\text{.}\)

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16.

Prove that if \(n\in\mathbb Z\text{,}\) then the sum \(n^3+(n+1)^3+(n+2)^3\) is divisible by \(9\text{.}\)

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17.

Prove that \(\forall a\in \mathbb Z\text{,}\) \(a^5\equiv a \pmod 5\text{.}\)

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18.

Without using the triangle inequality, prove that if \(x\in\mathbb{R}\text{,}\) then \(|x+4|+|x-3|\geq 7\text{.}\)

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19.

Let \(x\in\mathbb{R}\text{.}\) Show that if \(|x-1| \lt 1\text{,}\) then \(|x^2-1| \lt 3\text{.}\) You may use the following result without proof:

\begin{equation*} |ab|=|a|\cdot|b| \text{ for any } a,b\in\mathbb{R}. \end{equation*}

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20.

Let \(x\in\mathbb{R}\text{.}\) Show that if \(|x-2| \lt 1\text{,}\) then \(|2x^2-3x-2| \lt 7\text{.}\) You may use the following result without proof:

\begin{equation*} |ab|=|a|\cdot|b| \text{ for any } a,b\in\mathbb{R}. \end{equation*}

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21.

Prove the reverse triangle inequality. That is, given \(x, y\in \mathbb{R}\text{,}\) prove

\begin{equation*} |x-y|\geq \big||x|-|y|\big|. \end{equation*}

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22.

We say that a function \(f\) is decreasing on its domain \(D\) if for all \(x,y\in D\text{,}\) whenever \(x\leq y\text{,}\) we have \(f(x)\geq f(y)\text{.}\) Explain why the following statement is false:

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Let \(f:\mathbb{R}-\{0\}\to\mathbb{R}\) be defined by \(f(x)=1/x\text{.}\) Then \(f\) is decreasing.

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Rewrite the statement to make it true by changing the domain of the function \(f\text{.}\) Then prove your statement.

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