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Cheatsheets - Properties of Factors

Key Properties of Factors

  1. Definition: A factor of a number \(n\) is an integer that divides \(n\) without leaving a remainder.

    • Example: Factors of 12 are \(1, 2, 3, 4, 6, 12\).

  2. Factor Pairs: Factors come in pairs. If \(f\) is a factor of \(n\), then \(n/f\) is also a factor.

    • Example: Factor pairs of 24 are \((1, 24), (2, 12), (3, 8), (4, 6)\).

  3. 1 and the Number Itself: Every number has 1 and itself as factors.

    • Example: Factors of 7 include 1 and 7.

  4. Finite Set of Factors: A non-zero integer has a finite number of factors, determined by its prime factorization.

    • Example: \(36 = 2^2 \times 3^2\) has 9 factors.

  5. Symmetry in Factors: If \(a\) divides \(b\), then \(b/a\) is also a factor of \(b\), forming symmetrical pairs.

  6. Prime Numbers: Prime numbers have exactly two factors: 1 and the prime itself.

    • Example: The prime number 13 has factors 1 and 13.

  7. Composite Numbers: Composite numbers have more than two factors.

    • Example: 18 is composite, with factors \(1, 2, 3, 6, 9, 18\).

  8. Divisibility: Factors relate directly to divisibility. If \(a\) divides \(b\), \(a\) is a factor of \(b\).

    • Example: \(4 \div 2 = 2\), so 2 is a factor of 4.

  9. Greatest Common Factor (GCF): The GCF of two numbers is the largest factor they share.

    • Example: \(\text{gcf}(36, 48) = 12\).

  10. Negative Factors: Every positive factor has a corresponding negative factor.

    • Example: Factors of 12 include \(-1, -2, -3, -4, -6, -12\).

  11. Coprime (Relatively Prime) Numbers: Two or more numbers are coprime if their greatest common factor (GCF) is 1, meaning they share no factors other than 1.

    • Example: \(15\) and \(8\) are coprime because \(\text{gcf}(15, 8) = 1\).

These properties form the foundation of understanding number theory, divisibility, and the structure of integers.