1. Index Divisibility GCD and LCM Exercises
Factors and Multiples
Matem´ticas 2o E.S.O.
a
Alberto Pardo Milan´s
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2. Index Divisibility GCD and LCM Exercises
1 Divisibility
2 GCD and LCM
3 Exercises
Alberto Pardo Milan´s
e Factors and Multiples

3. Index Divisibility GCD and LCM Exercises
Divisibility
Alberto Pardo Milan´s
e Factors and Multiples

4. Index Divisibility GCD and LCM Exercises
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.
Alberto Pardo Milan´s
e Factors and Multiples

5. Index Divisibility GCD and LCM Exercises
Divisibility
Primes
A prime number is a positive number that has only two positive
factors 1 and the number itself (1 is not considered a prime
number as it only has one positive factor). A number with more
than 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 positive
factors (1, 2, 3 and 6).
Two numbers are relatively prime if they have no common positive
divisors except 1.
Example: 6 and 25 are relatively prime because the positive factors
of 6 are 1, 2, 3, 6 and the positive factors of 25 are 1, 5, 25.
Alberto Pardo Milan´s
e Factors and Multiples

6. Index Divisibility GCD and LCM Exercises
Divisibility
Prime decomposition
Prime decomposition is to find the set of prime factors of an
integer: To factorize a number you have to express the number as
a product of its prime factors.
To factorize negative numbers use also −1.
Examples:
45 3
15 3
⇒ 45 = 3 · 3 · 5 = 32 · 5.
5 5
1
25 5
5 5 ⇒ −25 = −1 · 5 · 5 = −1 · 52 .
1
Alberto Pardo Milan´s
e Factors and Multiples

7. Index Divisibility GCD and LCM Exercises
GCD and LCM
Alberto Pardo Milan´s
e Factors and Multiples

8. Index Divisibility GCD and LCM Exercises
GCD and LCM
The Greatest Common Divisor (GCD) is the highest number that
is a common factor of two or more numbers. It is clear that if
GCD(a, b) = 1, a and b are relatively prime.
Example: GCD(42, 110) = 2, because positive factors of 42
are 1, 2, 3, 6, 7, 14, 21, 42, and positive factors of 110 are
1, 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 prime
factors of these numbers.
650 = 2 · 5 · 5 · 13
Example:
440 = 2 · 2 · 2 · 5 · 11
=⇒GCD(440, 650) = 2 · 5 = 10
Alberto Pardo Milan´s
e Factors and Multiples

9. Index Divisibility GCD and LCM Exercises
GCD and LCM
The Least Common Multiple (LCM) is the lowest positive number
that is a common multiple of two or more numbers.
Example: LCM(6, 9) = 18, because positive multiples of 6 are
6, 12, 18, 24, . . . and positive multiples of 9 are 9, 18, 27, . . .
To find the LCM first find the prime factorization of each number
and write it in index form. Then the LCM will be the product of
the each prime factors with the greatest power.
84 = 22 · 3 · 7
Example: 198 = 2 · 32 · 11
2772 = 22 · 32 · 7 · 11
=⇒LCM(84, 198) = 2772
Alberto Pardo Milan´s
e Factors and Multiples

10. Index Divisibility GCD and LCM Exercises
Exercises
Alberto Pardo Milan´s
e Factors and Multiples

11. Index Divisibility GCD and LCM Exercises
Exercises
Exercise 1
There are 28 students in our class and we want to divide them into
groups with equal number of students. How many ways can the
class be divided into groups? What are the results?
Alberto Pardo Milan´s
e Factors and Multiples

12. Index Divisibility GCD and LCM Exercises
Exercises
Exercise 2
Mary wants to serve hotdogs for 48 people. Sausages come in
packages of 8 and hot dog buns come in packages of 12. She wants
to have enough to serve everyone and have no leftovers. How many
packages of sausages and hotdog buns should she purchase?
Alberto Pardo Milan´s
e Factors and Multiples

13. Index Divisibility GCD and LCM Exercises
Exercises
Exercise 3
Peter works in a florist shop. Today He has to make identical floral
arrangements for a bridal party. He has 84 daisies, 66 lilies, and 30
orchids. He wants each arrangement to have the same number of
each flower. What is the greatest number of arrangements that he
can make if every flower is to be used?
Alberto Pardo Milan´s
e Factors and Multiples

14. Index Divisibility GCD and LCM Exercises
Exercises
Exercise 4
Samantha loves the sea. She has kayaking lessons every fifth day
and diving lessons every seventh day. If she had a kayaking lesson
and a diving lesson on June the sixth, when will be the next date
on which she has both kayaking and diving lessons?
Alberto Pardo Milan´s
e Factors and Multiples

15. Index Divisibility GCD and LCM Exercises
Exercises
Exercise 5
There are two flashing neon lights. One blinks every 4 seconds and
the other blinks every 6 seconds. If they are turned on exactly at
the same time, how many times will they blink at the same time in
a minute?
Alberto Pardo Milan´s
e Factors and Multiples

16. Index Divisibility GCD and LCM Exercises
Exercises
Exercise 6
Peter sells books. He made 240e selling children’s books, 140e
from cookbooks, and 280e from paperback books. He gets exactly
the same benefit from each book. What is the most that Peter
could get for each book? How many books would Peter have sold
then?
Alberto Pardo Milan´s
e Factors and Multiples