# Factors and Multiples ```Summary sheet on the last page
FACTORS AND MULTIPLES – Answer Key
I. Find prime factors by factor tree method
a. 768
2 384
2
192
2
96
2
48
2
24
2
12
2
6
2
768 = 2*2*2*2*2*2*2*2 *3
b. 1608
3
536
2
268
2
134
2
67
1608 = 3*2*2*2*67
c. 2445
5
489
3
163
2445= 5*3*163
3
Summary sheet on the last page
II.
Find prime factors by prime factorization method
a.
3 2793
931
Ans = 3* 931
b.
5 595
7 119
17
Ans = 5*7* 17
c.
7 119
17
Ans = 7*17
III.
Find if the following are coprime or not. If they are not coprime find a common factor.
a. 486, 81 = These numbers are not coprime as they have a common factor which is 3.(3
divides both 486,81 without a remainder)
b. 49,59 = These numbers are coprime as they don’t have a common factor which divides
both numbers without a remainder
c. 75,101 = These numbers are coprime as they don’t have a common factor which divides
both numbers without a remainder
d. 144,192 = These numbers are not coprime as they have two common factors which are 2,
3 . (2 and 3 divides both 144,192 without a remainder)
e. 246,780 = These numbers are not coprime as they have a common factor which is 2.(2
divides both 246,780 without a remainder)
Summary sheet on the last page
f. 634,597 = These numbers are coprime as they don’t have common factor which divides
both numbers without a remainder
IV.
Find the LCM of
a.
5 175, 80, 25
5 35, 16, 5
2 7 , 16,
2 7, 8,
2 7, 4,
* 7, 2,
1
1
1
1
Ans = 5*5*2*2*2*7*2 = 2800
b.
2 288, 200, 64
2 144, 100, 32
2 72, 50, 16
2 36, 25, 8
2 18, 25, 4
3 9, 25, 2
5 3, 25, 2
3,
5, 2
*
Ans = 2*2*2*2*2*3*5*3*5*2 = 14400
c. 5
3
3
2
2
2
*
135,
27,
9,
3,
3,
3,
3,
48,
48,
16,
16,
8,
4,
2,
25
5
5
5
5
5
5
Summary sheet on the last page
Ans = 5*3*3*2*2*2*3*2*5 = 10,800
d.
3 168, 66, 12
2 56, 22, 4
2 28, 11, 2
2 14, 11, 1
7, 11, 1
Ans = 3*2*2*2*7*11*1 = 1848
e. 5
5
2
7
550,
110,
22,
11,
11,
25,
5,
1,
1,
1,
140
28
28
14
2
*
Ans = 5*5*2*7*11*2 = 7700
f. 5 420, 330,
2 84, 66,
7 42, 33,
3 6, 33,
2 2, 11,
1, 11,
*
280
56
28
4
4
2
Ans = 5*2*7*3*2*11*2 = 9240
g. 5 65, 52
13 13, 52
2 1, 4
1, 2
Ans = 5*13*2*2= 260
Summary sheet on the last page
V.
Find the HCF by prime factorization method
a. 3 27, 36, 18
3 9, 12, 6
3, 4, 2
Ans = 3*3 = 9
b. 5 25, 95, 80
5 5, 19, 16
Ans = 5*5 = 25
c. 5 750, 500, 250
5
150, 100, 50
5
2
30, 20, 10
6, 4, 2
3, 2, 1
Ans = 5*5*5*2 = 250
d.
2 252, 126
3
3
126,
42,
7 14,
2,
63
21
7
1
Ans =2*3*3*7 = 126
e. 2 56, 144, 108
2 28, 72,
14,
54
36, 27
Summary sheet on the last page
Ans =2*2 = 4
VI.
Find the HCF by successive division method
a. 106,192,96
_________
106 ) 192(1
-106________
86 ) 106( 1
-86____
20) 86(4
-80_____
6 ) 20(3
-18_____
2) 6(3
-6______
0______
HCF of 106 and 192 is 2
________
2)96(48
_-96_
0__
Ans - HCF of 106,192 and 96 is 2
b. 108,144,60
____________
108) 144(1
_-108______
36) 108(3
_-108___
0__
So, HCF of 108 and 144 is 36
Summary sheet on the last page
________
36) 60(1
-36_____
24) 36(1
_-24_____
12)24(2
-24___
0____
Ans – HCF of 108,144,60 is 12
c.
7, 14,24
___________
7) 14 (2
-14
0
So, HCF of 7,14 is 7
_____
7) 24(4
-24__
0__
Ans – HCF of 7,14,24 is 7
d. 32,56,46
______
32)56(1
-32______
24 ) 32(1
-24____
8 ) 24(3
-24__
0__
Summary sheet on the last page
So HCF of 32 and 56 is 8
_________
8) 46(5
-40_____
6) 8 (1
-6____
2)6(3
-6___
0___
Ans = HCF of 32,56,46 is 2
VII.
Fill in the blanks
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
Every number is a factor of itself
The only even prime is 2
All even numbers are multiples of 2
The 12th multiple of 5 is 60
Numbers which have only 1 and itself as factors are called prime numbers
Numbers which have more than one factor are called composite numbers
Two numbers are coprime if they don’t have a common factor
Factors of 81 are 9 and 9
Set of whole numbers is { 0,1,2,3,4…..}
563,291,497,893 are odd numbers
Two numbers are not coprime if they have a common factor
Multiple of 7 and 8 is 56
The smallest multiple of every number is itself
A number that divides another number without a remainder is
called its factor
***************** THE END*********************
Summary sheet on the last page
Summary – Factors and Multiples
A number which divides another number without leaving a remainder is called its factor.
Eg. 3 divides 27 without a remainder . So 3 is a factor of 27
A number is a multiple of another number if it is divisible by it
Set of Natural numbers { 1,2,3,4…….}
Set of Whole numbers { 0,1,2,3,4……..}
Set of Integers {-3,-2,-1,0,1,2,3………}
Even numbers end with 0,2,4,6,8 and all even numbers are multiples of 2
Odd numbers end with 1,3,5,7,9
A number which had only one and itself as its factors is called prime numbers
Numbers which have more than one factor is called composite
1 – is unique and is neither prime nor composite
Only even prime is 2
Two numbers which have no factors in common are called coprime
Two numbers which have factors in common are not coprime
Summary sheet on the last page
Summary – HCF and LCM
1. Finding factors by factor tree method
Since it is called factor tree, use the tree method(this clue will help you not to get confused with
the prime factorization method since it looks like a tree)
Divide the given number using only prime numbers till you get two prime numbers in the end
Then answer should be written as “Given number = Product of all the prime factors at each
step”
eg. 1608
3
Use only prime numbers
536
2
268
2
134
2
67
2. Finding factors by prime factorization
Divide the number using only prime numbers till you cannot divide any further
Answer should be written as Product of all the factors and the numbers in the last row ”
Use only prime numbers
5 595
7 119
17
Ans = 5*7* 17
3. LCM
Divide all the given numbers using only prime numbers.
You can use prime numbers that need not divide all the given numbers. In that
case, carry down the numbers which are not divisible as it is
Continue till you can’t divide further
Answer = Product of all factors and the numbers on the last row
5 65, 52
13 13, 52
2 1, 4
1, 2
Number brought down as it is since
it is not divisible by 5
Always use prime numbers
Summary sheet on the last page
Ans = 5*13*2*2= 260
4. HCF – Prime Factorization method
Divide all the given numbers using a prime number . Note that here the prime number
must divide all the given numbers unlike LCM
Continue till you can’t find a prime number that will divide all the numbers
Answer = Product of all factors
2 56, 144, 108
2 28, 72,
14,
54
36, 27
Stop division since there is no prime
which divides all 3 numbers
Ans =2*2 = 4
Note: In order to differentiate between LCM and HCF just remember
In LCM take the factors on the vertical and horizontal (like in the letter L).
L
In HCF take only the factors in the vertical
In LCM, prime numbers need not divide all given numbers but HCF it must divide all the
numbers
5. HCF – Successive division method
Eg – HCF of 108,144 and 60
____________
108) 144(1
Remainder
_-108______
becomes
36) 108(3
second divisor
_-108___
0__
Divide second number by the first
This number becomes the second dividend
Continue the same process till you get zero
The last divisor is the HCF for the first two numbers
Summary sheet on the last page
Then divide the third number with this HCF in the same manner till you get zero. The last divisor
is the HCF of all 3 numbers
________Divide the third number by the HCF of the first numbers
36) 60(1
-36_____
24) 36(1
__-24_____
12)24(2
-24___
0____
Last divisor is the HCF of all three numbers
Ans – HCF of 108,144,60 is 12
******************** THE END***********************
```