Divisibility and GCF Quiz

Divisibility

• Definition: A number is divisible by another if the division results in an integer with no remainder.
• Basic Rules:
  • 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
  • 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • 5: A number is divisible by 5 if its last digit is 0 or 5.
  • 10: A number is divisible by 10 if its last digit is 0.
  • 4: A number is divisible by 4 if the last two digits form a number that is divisible by 4.
  • 6: A number is divisible by 6 if it is divisible by both 2 and 3.
  • 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

Writing Expressions by GCF (Greatest Common Factor)

• Definition: GCF is the largest number that divides two or more numbers without leaving a remainder.

• Finding GCF:

  • List Factors: List all the factors of each number and identify the largest common factor.
  • Prime Factorization: Break down each number into its prime factors and multiply the smallest power of all common primes.

• Expressing Numbers using GCF:

  • If two numbers, ( a ) and ( b ), share a GCF of ( g ), they can be expressed as:
    • ( a = g \times m )
    • ( b = g \times n )
  • Where ( m ) and ( n ) are co-prime (having no common factors other than 1).

• Examples:

  • For numbers 24 and 36:
    • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
    • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
    • GCF = 12
    • ( 24 = 12 \times 2 ), ( 36 = 12 \times 3 )

Practical Applications

• Divisibility Tests: Helpful in simplifying fractions or finding common denominators.
• GCF in Algebra: Useful in factoring polynomials and simplifying expressions.

Divisibility

• A number is divisible by another if division yields an integer with no remainder.
• To check divisibility by 2, the last digit must be even (0, 2, 4, 6, 8).
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• For divisibility by 5, the last digit must be either 0 or 5.
• A number is divisible by 10 if its last digit is 0.
• To determine divisibility by 4, the number formed by the last two digits must be divisible by 4.
• A number is divisible by 6 if it meets the criteria for both 2 and 3.
• A number is divisible by 9 if the sum of its digits is divisible by 9.

Writing Expressions by GCF (Greatest Common Factor)

• GCF is the largest number that divides two or more numbers without leaving a remainder.
• To find GCF, list all factors of each number and identify the largest common factor.
• Another method for finding GCF is through prime factorization; break down each number into its prime factors and multiply the smallest power of the common primes.
• Numbers ( a ) and ( b ) sharing a GCF of ( g ) can be expressed as ( a = g \times m ) and ( b = g \times n ), where ( m ) and ( n ) are co-prime.
• For the numbers 24 and 36, the factors are:
  • 24: 1, 2, 3, 4, 6, 8, 12, 24
  • 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• The GCF of 24 and 36 is 12, leading to ( 24 = 12 \times 2 ) and ( 36 = 12 \times 3 ).

Practical Applications

• Divisibility tests are useful for simplifying fractions and identifying common denominators.
• GCF plays a significant role in algebra, particularly in factoring polynomials and simplifying expressions.

Test your knowledge on divisibility rules and how to find the Greatest Common Factor (GCF) of numbers. This quiz covers essential concepts and techniques for determining divisibility by key numbers and identifying the largest common factor. Perfect for math students looking to reinforce their understanding.