Factors and multiples

Index Divisibility GCD and LCM Exercises

Factors and Multiples

Matematicas 2o E.S.O.Alberto Pardo Milanes

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1 Divisibility

2 GCD and LCM

3 Exercises

Alberto Pardo Milanes Factors and Multiples


Divisibility



Divisibility

Factors and multiples

A factor of a number n, is a number d which divides n.Read ⇐⇒ if and only if.d is a factor of n ⇐⇒d is a divisor of n ⇐⇒d divides n ⇐⇒n is divisible by d ⇐⇒n is a multiple of d.

Examples:−7 divides 14 ⇐⇒−7 is a factor of 14 ⇐⇒14 is divisible by −7 ⇐⇒14 is a multiple of −7.



Divisibility

Primes

A prime number is a positive number that has only two positivefactors 1 and the number itself (1 is not considered a primenumber as it only has one positive factor). A number with morethan two positive factors is called a composite number.

Examples: 3 is a prime number because has only two positive fac-tors (1 and 3). 6 is a composite number because has four positivefactors (1, 2, 3 and 6).

Two numbers are relatively prime if they have no common positivedivisors except 1.

Example: 6 and 25 are relatively prime because the positive factorsof 6 are 1, 2, 3, 6 and the positive factors of 25 are 1, 5, 25.



Divisibility

Prime decomposition

Prime decomposition is to find the set of prime factors of aninteger: To factorize a number you have to express the number asa product of its prime factors.To factorize negative numbers use also −1.Examples:

45 315 35 51

⇒ 45 = 3 · 3 · 5 = 32 · 5.

25 55 51

⇒ −25 = −1 · 5 · 5 = −1 · 52.



GCD and LCM



GCD and LCM

The Greatest Common Divisor (GCD) is the highest number thatis a common factor of two or more numbers. It is clear that ifGCD(a, b) = 1, a and b are relatively prime.

Example: GCD(42, 110) = 2, because positive factors of 42are 1, 2, 3, 6, 7, 14, 21, 42, and positive factors of 110 are1, 2, 5, 10, 11, 22, 55, 110.12 and 35 are relatively prime, because GCD(12, 35) = 1.

To find the GCD first find the prime factorization of each number.Then the GCD is the number that contains all the common primefactors of these numbers.

Example:650 = 2 · 5 · 5 · 13440 = 2 · 2 · 2 · 5 · 11

=⇒GCD(440, 650) = 2 · 5 = 10



GCD and LCM

The Least Common Multiple (LCM) is the lowest positive numberthat is a common multiple of two or more numbers.

Example: LCM(6, 9) = 18, because positive multiples of 6 are6, 12, 18, 24, . . . and positive multiples of 9 are 9, 18, 27, . . .

To find the LCM first find the prime factorization of each numberand write it in index form. Then the LCM will be the product ofthe each prime factors with the greatest power.

Example:

84 = 22 · 3 · 7198 = 2 · 32 · 11

2772 = 22 · 32 · 7 · 11=⇒LCM(84, 198) = 2772



Exercises



Exercises

Exercise 1

There are 28 students in our class and we want to divide them intogroups with equal number of students. How many ways can theclass be divided into groups? What are the results?



Exercises

Exercise 2

Mary wants to serve hotdogs for 48 people. Sausages come inpackages of 8 and hot dog buns come in packages of 12. She wantsto have enough to serve everyone and have no leftovers. How manypackages of sausages and hotdog buns should she purchase?



Exercises

Exercise 3

Peter works in a florist shop. Today He has to make identical floralarrangements for a bridal party. He has 84 daisies, 66 lilies, and 30orchids. He wants each arrangement to have the same number ofeach flower. What is the greatest number of arrangements that hecan make if every flower is to be used?



Exercises

Exercise 4

Samantha loves the sea. She has kayaking lessons every fifth dayand diving lessons every seventh day. If she had a kayaking lessonand a diving lesson on June the sixth, when will be the next dateon which she has both kayaking and diving lessons?



Exercises

Exercise 5

There are two flashing neon lights. One blinks every 4 seconds andthe other blinks every 6 seconds. If they are turned on exactly atthe same time, how many times will they blink at the same time ina minute?



Exercises

Exercise 6

Peter sells books. He made 240e selling children’s books, 140efrom cookbooks, and 280e from paperback books. He gets exactlythe same benefit from each book. What is the most that Petercould get for each book? How many books would Peter have soldthen?


Education

Factors and multiples