Factors Multiples and Primes

Learn greatest common factor, least common multiple, primes and composite numbers.

Lesson 2 of 33 parts Practice Flashcards

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Greatest Common Factor

Objective: Explain greatest common factor, apply the stated method, and check its conditions.

A factor divides an integer without remainder. The greatest common factor is the largest positive integer dividing both given positive integers. Prime factorization finds it by retaining shared prime powers with the smaller exponent. A common factor divides both numbers exactly.

Formulas

GCF uses shared prime powers with minimum exponents

Worked examples

Try itFind the greatest common factor of 42 and 49.Show answer

Step 1: Factor or list divisors of 42 and 49. Step 2: Keep divisors that occur in both lists. Step 3: The greatest is 7.

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Greatest Common Factor

Objective: Explain greatest common factor, apply the stated method, and check its conditions.

Formulas

GCF uses shared prime powers with minimum exponents

Worked examples

Try itFind the greatest common factor of 42 and 49.Show answer

Step 1: Factor or list divisors of 42 and 49. Step 2: Keep divisors that occur in both lists. Step 3: The greatest is 7.

Least Common Multiple

Objective: Explain least common multiple, apply the stated method, and check its conditions.

The least common multiple is the smallest positive integer divisible by both given positive integers. Prime factorization retains every required prime power with the larger exponent. It is useful for matching denominators or recurring schedules. The least common multiple is positive; zero is not used.

Formulas

$LCM(a,b)=ab/GCF(a,b), a,b>0$

Worked examples

Try itFind the least common multiple of 6 and 7.Show answer

Step 1: Use LCM(a,b)=|ab|/GCF(a,b). Step 2: The GCF is 1. Step 3: The LCM is 42.

Primes and Composite Numbers

Objective: Explain primes and composite numbers, apply the stated method, and check its conditions.

A prime is an integer greater than one with exactly two positive factors, one and itself. A composite integer has more than two positive factors. One is neither prime nor composite. Factor lists and divisibility checks classify a candidate. One is neither prime nor composite.

Formulas

Prime n>1 has exactly two positive factors

Worked examples

Try itHow many positive factors does 8 have?Show answer

Step 1: Find divisors of 8 that leave no remainder. Step 2: The positive factors are 1, 2, 4, 8. Step 3: There are 4; a prime has exactly two positive factors.

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