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Mathematics LibreTexts

7.5: Exercise

  • Page ID
    24839
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    Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4–7.

    Exercise \(\PageIndex{1}\)

    Suppose \(x \in \mathbb{Z}\). Then x is even if and only if \(3x+5\) is odd.

    Exercise \(\PageIndex{2}\)

    Suppose \(x \in \mathbb{Z}\). Then x is odd if and only if \(3x+6\) is odd.

    Exercise \(\PageIndex{3}\)

    Given an integer a, then \(a^3+a^2+a\) is even if and only if a is even.

    Exercise \(\PageIndex{4}\)

    Given an integer a, then \(a^2+4a+5\) is odd if and only if a is even.

    Exercise \(\PageIndex{5}\)

    An integer a is odd if and only if \(a^3\) is odd.

    Exercise \(\PageIndex{6}\)

    Suppose \(x, y \in \mathbb{R}\).Then \(x^3+x^{2}y = y^2+xy\) if and only if \(y = x^2\) or \(y = -x\).

    Exercise \(\PageIndex{7}\)

    Suppose \(x, y \in \mathbb{R}\). Then \((x+y)^2 = x^2+y^2\) if and only if \(x = 0\) or \(y = 0\).

    Exercise \(\PageIndex{8}\)

    Suppose \(a, b \in \mathbb{Z}\). Prove that \(a \equiv b \pmod{10}\) if and only if \(a \not\equiv b \pmod{2}\) and \(a \not\equiv b \pmod{5}\).

    Exercise \(\PageIndex{9}\)

    Suppose \(a \in \mathbb{Z}\). Prove that \(14|a\) if and only if \(7|a\) and \(2|a\).

    Exercise \(\PageIndex{10}\)

    If \(a \in \mathbb{Z}\), then \(a^3 \equiv a \pmod{3}\).

    Exercise \(\PageIndex{11}\)

    Suppose \(a, b \in \mathbb{Z}\). Prove that \((a-3)b^2\) is even if and only if a is odd or b is even.

    Exercise \(\PageIndex{12}\)

    There exists a positive real number x for which \(x^2 < \sqrt{x}\).

    Exercise \(\PageIndex{13}\)

    Suppose \(a, b \in \mathbb{Z}\). If \(a+b\) is odd, then \(a^2+b^2\) is odd.

    Exercise \(\PageIndex{14}\)

    Suppose \(a \in \mathbb{Z}\). Then \(a^2|a\) if and only if \(a \in \{-1,0,1\}\).

    Exercise \(\PageIndex{15}\)

    Suppose \((a, b \in \mathbb{Z}\). Prove that \(a+b\) is even if and only if a and b have the same parity.

    Exercise \(\PageIndex{16}\)

    Suppose \(a,b \in \mathbb{Z}\). If ab is odd, then \(a^2+b^2\) is even.

    Exercise \(\PageIndex{17}\)

    There is a prime number between 90 and 100.

    Exercise \(\PageIndex{18}\)

    There is a set X for which \(\mathbb{N} \in X\) and \(\mathbb{N} \subseteq X\).

    Exercise \(\PageIndex{19}\)

    If \(n \in \mathbb{N}\), then \(2^0+2^1+2^2+2^3+2^4+ \cdots +2^n = 2^{n+1}-1\).

    Exercise \(\PageIndex{20}\)

    There exists an \(n \in \mathbb{N}\) for which \(11|(2n-1)\).

    Exercise \(\PageIndex{21}\)

    Every real solution of \(x^3+x+3 = 0\) is irrational.

    Exercise \(\PageIndex{22}\)

    If \(n \in \mathbb{Z}\), then \(4|n^2\) or \(4|(n^2-1)\).

    Exercise \(\PageIndex{23}\)

    Suppose a, b and c are integers. If \(a|b\) and \(a|(b^2-c)\), then \(a|c\).

    Exercise \(\PageIndex{24}\)

    If \(a \in \mathbb{Z}\), then \(4 \nmid (a^2-3)\).

    Exercise \(\PageIndex{25}\)

    If \(p > 1\) is an integer and \(n \nmid p\) for each integer n for which \(2 \le n \le \sqrt{p}\), then p is prime.

    Exercise \(\PageIndex{26}\)

    The product of any n consecutive positive integers is divisible by n!.

    Exercise \(\PageIndex{27}\)

    Suppose \(a, b \in \mathbb{Z}\). If \(a^2+b^2\) is a perfect square, then a and b are not both odd.

    Exercise \(\PageIndex{28}\)

    Prove the division algorithm: If \(a, b \in \mathbb{N}\), there exist unique integers q, r for which \(a = bq+r\), and \(0 \le r < b\). (A proof of existence is given in Section 1.9, but uniqueness needs to be established too.)

    Exercise \(\PageIndex{29}\)

    If \(a|bc\) and \(gcd(a, b) = 1\), then \(a|c\). (Suggestion: Use the proposition on page 152.)

    Exercise \(\PageIndex{30}\)

    Suppose \(a, b, p \in \mathbb{Z}\) and p is prime. Prove that if \(p|ab\) then \(p|a\) or \(p|b\). (Suggestion: Use the proposition on page 152.)

    Exercise \(\PageIndex{31}\)

    If \(n \in \mathbb{Z}\), then \(gcd(n, n+1) = 1\).

    Exercise \(\PageIndex{32}\)

    If \(n \in \mathbb{Z}\), then \(gcd(n, n+2) \in \{1, 2\}\).

    Exercise \(\PageIndex{33}\)

    If \(n \in \mathbb{Z}\), then \(gcd(2n+1, 4n^2+1) = 1\).

    Exercise \(\PageIndex{34}\)

    If \(gcd(a, c) = gcd(b, c) = 1\), then \(gcd(ab, c) = 1\). (Suggestion: Use the proposition on page 152.)

    Exercise \(\PageIndex{35}\)

    Suppose \(a, b \in \mathbb{N}\). Then \(a = gcd(a,b)\) if and only if \(a|b\).

    Exercise \(\PageIndex{36}\)

    Suppose \(a, b \in \mathbb{N}\). Then \(a = lcm(a,b)\) if and only if \(b|a\).