Factors and Multiples

1. Factors and Multiples

2. Definition of Factors and Multiples • If one number is a factor of a second number or divides the second (as 3 is a factor of 12), then the second number is a multiple of the first (as 12 is a multiple of 3).

3. Definition of Factors and Multiples • Also, if a and b are whole numbers and a does not equal 0, then a is a factor of b if and only if there is a whole number c such that ac = b.

4. Components of Factors and Multiples • Divisibility Tests • Prime Numbers • Greatest Common Factor • Least Common Multiple

5. Divisibility Tests A quick way to see if numbers are divisible

6. Divisibility by 2 or 5 • A number is divisible by 2 if the number represented by the units digit is divisible by 2. The number is divisible by 2 if the units digit is 0, 2, 4, 6, or 8. • A number is divisible by 5 if the number represented by the units digit is divisible by 5. A number is divisible by 5 if its unit digit is 0 or 5.

7. Divisibility by 3 or 9 • A number is divisible by 3 if the sum of its digits is divisible by 3. • A number is divisible by 9 if the sum of its digits is divisible by 9.

8. Divisibility by 4 or 6 • A number is divisible if the number represented by the last two digits of a number is divisible by 4, then the original number will be divisible by 4. • A number is divisible by both 2 or 3, then it is divisible by 6. If it is not divisible by 2 and 3, then it is not divisible by 6.

9. Prime Numbers

10. Definition of a Prime Number • A number which is divisible by 1 and itself, only has 2 factors. • The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

11. Prime Number Test • Suppose n is a whole number and k is the smallest whole number such that k * k is greater than n. If there is no prime number less than k that is a factor of n, then n is a prime number.

12. Greatest Common Factor

13. Definition of Greatest Common Factor • For any two nonzero whole numbers a and b, the greatest common factor, written GCF(a, b) is the greatest factor (divisor) of both a and b.

14. How to find Greatest Common Factor • To find the GCF of 24 and 36: • List all factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. • List all factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36. • The common factors are 1, 2, 3, 4, 6, and 12. The greatest (largest) is 12, therefore 12 is the greatest common factor of 24 and 36.

15. Least Common Multiple

16. Definition of Least Common Multiple • A number is called a common multiple of two numbers if it is a multiple of both. • Common multiples of 5 and 7 are: 35, 70, 105, 140, 175, etc. • Because 35 is the smallest common multiple it is known as the least common multiple. • In other words, for any two nonzero whole numbers a and b, the least common multiple written LCM(a, b), is the smallest multiple of both a and b.

17. How to find Least Common Multiple • For positive integers a and b, LCM(a,b): (a * b)/(GCF(a,b)) In other words, multiply the two numbers you are finding the least common multiple of and divide that answer by the greatest common factor of the two numbers. Also, when GCF(a, b) = 1, LCM(a, b) = a * b.

18. Find LCM (28, 44) • 28 *44 = 1232 • GCF (28, 44) = 4 • 1232/4 = 308 • 308 is the Least Common Multiple of 28 and 44.

19. Now go back to the lesson home page • When doing the assignment, you can use this presentation to help guide you through. • When doing the quiz, use nothing but your pencil and calculator, do not use this presentation.