Download UNIT 3: Divisibility in Natural Numbers 3.1 Relationship of divisibility

UNIT 3: Divisibility in Natural Numbers
3.1 Relationship of divisibility
In a division of natural numbers, we can find four elements: D=dividend, d=divisor,
q=quotient and r=remainder.
Dividend (D)
divisor (d)
quotient (q)
remainder (r)
A division is exact, if its remainder is zero. In this case, D=d·q is verified.
A division isn't exact, if its remainder isn't zero. In this case, D=d·q+r is verified.
When the division between two numbers is exact, we say there is a relationship of
divisibility between them.
D is divisible by d if r = 0
Example: Is there a relationship of divisibility between 78 and 6?
78
6
* The division 78:6 is ………, because the remainder is zero.
* 78 is divisible by 6.
* ….. is a factor of 78
* 6 is a ………of 78
There is a relationship of divisibility between 78 and 6.
Exercises:
1.Is 124 divisible by any of these numbers? Justify your answer.
a) 4
b) 3
124 is……………by 4
…..and……are factors of 124
124 is………………..by 3
……..and ……are factors of………
2. Is there a relationship of divisibility between these couples of numbers?
a) 322 and 14
b) 894 and 6
c) 135 and 7
3.2 Multiples of a number
A number b is a multiple of another number a if the division b:a is exact
Example: 36 is a multiple of 4 but it isn't a multiple of 5.
The multiple of a number is the product generated when that number is multiplied by a
natural number.
Multiples of 4 = 4·1, 4·2, 4·3, 4·4, 4·5, …
And we write it like this:
4 = 4, 8, 12, 16, 20, ...
Every numbers is a multiple of itself and of 1
Exercises
1.Calculate the first eight multiples of number 3
3 
2. Which of these numbers are multiples of number 8?
12, 36,19, 72,56,23,32
3.3 Test of divisibility
Here are some quick and easy checks to see if one number will divide exactly:
Divisible by 2
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8
Example: 2 346 is divisible by 2 because the last digit is 6.
Divisible by 3
A number is divisible by 3 if the sum of the digits is divisible by 3.
Example: 23 457 is divisible by 3 because the sum of the digits is 21 (2+3+4+5+7=21), and
21 is divisible by 3.
Divisible by 5
A number is divisible by 5 if the last digit is either 0 or 5.
Example: 9 876 345 is divisible by 5 because the last digit is 5
Exercise: Which of the following numbers are multiple of number 3? And of number 5?
23,45,69,13,18,66 and 90
3.4 Prime and composite numbers:
A prime number (or a prime) is a natural number which has exactly two different divisors:
1 and itself. Ex: 2,3,5…..
If a number has more than two divisors, it is called composite number.Ex: 4, 6,30…..
Number 1 is not a prime number.
Exercise: Write the first ten prime numbers:
Exercise: Separate prime and composite numbers: 2, 6, 5, 94, 38, 37, 23, 51
3.5 Factoring numbers
Decompose a number in prime factors (prime factoring or prime factor decomposition of
a number) consists of dividing the number between its prime factors until obtaining
number 1 in the quotient.
Example:
24 2 ← 24 is divisible by 2
12 2 ← 12 is divisible by 2
6 2 ← 6 is divisible by 2
3 3 ← 3 is divisible by 3
1
24 =
Exercise: Find the prime factor decomposition of the following numbers:
28
36
81
3.6. Lowest Common Multiple (L.C.M )
Method 1
The first few multiples of 4 are 4, , 12, , , 24, 28,……………………
The first few multiples of 5 are , 10, 15,
, 25, , ……………………
The L.C.M of 4 and 5 is 20 because it is the lowest number which is in both lists.
Method 2
To work out the LCM of several numbers, first write them as a product of their
primes factors and then take the common and non-common factors with the highest
exponent
Example: Write down the L.C.M of 75 and 90
75
90
L.C.M (75,90) =
3.7 The Highest Common Factor ( H.C.F)
Method 1
The factors of 12 are 1, 2, , 4, , 6, 12.
The factors of 20 are 1, 2, , , 10,
.
The H.C.F of 12 and 20 is 4 because it is the highest number which is in both lists.
Method 2
To work out the H.C.F. of several numbers, first write them as a product of their
primes factors and then take only the common factors with the least exponent
Example: Find the H.C.F of 40 and 60
40
60
EXERCISES
1. Is 250 divisible by any of these numbers? Justify your answer.
a) 2
d) 10
b) 3
e) 20
c) 5
f) 250
2. Is there a relationship of divisibility between these couples of numbers?
a) 224 and 16
d) 513 and 19
g) 20 and 300
b) 420 and 35
e) 688 and 44
h) 15 and 75
c) 613 and 13
f) 2070 and 46
i) 23 and 203
3. Find at least, 4 couples of numbers which have a relationship of divisibility:
420
9
13
18
70
156
9
6
11
4. Find all the factors of:
d) 10
c) 20
e) 30
g) 45
i) 8
k) 15 m) 55
e) 18
d) 24
f) 39
h) 50 j) 12
l) 28
n) 36
5. Which of these numbers are multiples of number 6?
10
12 16
30
42 54
60 76
90 148 174
6. Write down every factor of each number:
a)7
c) 17
e) 11
b)13
d) 29
f) 3
7. Which of these numbers are multiples of 2?
57 66 71 90
99 111 162
222
483
805
8.Write down the numbers which are multiples of 3.
1.023
754 679
555 510
390
186
173
9. Write down the numbers which are multiples of 5.
327 155
207
735
420 815
553
10. Find the L.C.M of:
a) L.C.M .( 10,15)
b) L.C.M.( 75, 90)
c) L.C.M.( 8, 12)
11. Find the L.C.M of:
a) L.C.M.( 18, 24, 30)
c) L.C.M. ( 30,40)
d) L.C.M.( 50, 75)
e) L.C.M.( 200, 300)
f) L.C.M.( 72, 108)
g) L.C.M.( 216,288)
h) L.C.M.( 36, 54)
b) L.C.M. (20, 30, 40)
12. Is the following sentence true?
15 is a multiple of 3 = 3 is a factor of 15 = 15 is divisible by 3?
13. Three clocks ring once at the same time. The first clock rings every 90 minutes, the
second one every 30 minutes, and the third one every 60 minutes. How many minutes will it
take until they ring together again?
14. Joan plays tennis every 4th day and Laura does it every 7th day. If both of them have
played today, when will both of them play together again?
15. A fruit company sends goods to Toledo every six days and to Lugo every 8 days.
Today both shipments have coincided. When will be next time they coincide?
16. The green line bus passes in front of my house every 20 minutes and the blue line
bus every 30 minutes. At 3 o’clock they passed together. At what time will they pass
together again?
17. Three sisters visit their grandmother that lives in Ferreries. Maria, the oldest
sister goes every 5 days. Teresa goes every 6 days, and the youngest sister called Lola
goes every 10 days. If today the three of them were together in her grandmother’s, when
will the next time they coincide be?
18.Work out:
a) H.C.F. (50, 75) d) H.C.F.( 120,144)
g) H.C.F. ( 56, 70) j)H (18,24 ,30)
b) H.C.F. ( 20, 24)
e) H.C.F.( 168, 196) h) H.C.F. (140, 180) k)H.C.F.(2,4,8)
c) H.C.F. ( 54, 60)
f) H.C.F. ( 24,36)
i) H.C.F. ( 180, 270)
19. Paul has three pieces of rope with lengths of 140 cm, 168 cm and 210 cm. He would
like to cut the three pieces of rope into smaller pieces of equal length with no
remainders. What is the longest possible length of the smallest pieces of rope?
20. There are 100 senators and 435 representatives in the United States of America
Congress. How many identical groups could be formed from all senators and
representatives (with the same number of senators and representatives in each group)
21. A choir director of your school wants to divide the choir into smaller groups.
There are 24 sopranos, 60 altos and 36 tenors. Each group will have the same number
of each type of voice.
a) What is the greatest number of groups that can be formed?
22. We have 80 litres of white wine and 60 litres of red wine. We want to package
them in bottles. Each bottle will contain the same amount of wine and we cannot mix
both types of wine. Which is the capacity of each bottle?
Check yourself:
1. Is there a relationship of divisibility between these couples of numbers?
a) 24 and 8
b) 365 and 15
c) 128 and 12
2. Write down the numbers which are multiples of 3
34, 54,98, 97, 23, 197, 286, 33
3. Point out which of these numbers are prime numbers:
1,5,4, 2, 24, 23, 76, 77, 91, 65.
4. Write down the L.C.M and the H.C.F of:
a) 20, 40, 50
b) 36,60,72
c) 22, 56, 8
5. Two sisters visit their grandmother that lives in Alaior. Susan goes every 12 days and
Teresa goes every 7 days. If today both of them were together in her grandmother’s,
when will the next time they coincide be?
6. We have 80 red balls and 60 blue balls. We want to package them in boxes. Each box will
contain the same amount of balls and we cannot mix both types of balls. Which is the
capacity of each box?