Question

1. Find the HCF of the given numbers by prime factorisation method:
(i) 28, 36
(ii) 54, 72,90 (iii) 105, 140, 175.
2. Find the HCF of the given numbers by division method:
(i) 198, 429
(ii) 20, 64, 104 (iii) 120, 144, 204.​

Answer 1

[ HINT: To find the HCF of the given number, then take the common factor and take its lowest power. ]

1. Finding the HCF bye prime factorization:

(i) 28, 36

So, Prime factorization of

28 = 2 *  2 * 7 = 2² × 7

36 = 2 * 2 * 3 * 3 = 2² × 3²

Now,

HCF(28,36) = 2² = 4

- - - - - -

(ii) 54, 72,90

54 = 2 * 3 * 3 * 3 = 2 × 3³

72 = 2 * 2 * 2 * 3 * 3 = 2³ × 3²

90 = 2 * 3 * 3 * 5 = 2 × 3² × 5

So,

HCF(54,72,90) = 2 × 3² = 2×9 = 18

- - - - -

(iii) 105, 140, 175

So, Prime factorization of

105 = 3 * 5 * 7

140 =  2 * 2 * 5 * 7 = 2² × 5 × 7

175 = 5 * 5 * 7 = 5² × 7

Now,

HCF(105, 140, 175) = 5 × 7 = 35

- - -- - - - -

2. Finding the HCF of given number by division method.

(i) 198, 429

Here,

429 = 198 * 2 + 33

198 = 33 * 5 + 33

33 = 33 * 1 + 0

So,

The HCF(198, 429) = 33

- - -

(ii) 20, 64, 104

Here,

64 = 20 * 3 + 4

20 = 4 * 5 + 0

Here we got last divisor = 4

104 = 4 * 26 + 0

So,

The Last divisor is 4

Hence,

HCF(20, 64, 104) = 4

- - - - - -

(iii) 120, 144, 204

Here,

144 = 120 * 1 + 24

120 = 24 * 5 + 0

Here the last divisor is 24

204 = 24 * 8 + 12

24 = 12 * 2 + 0

Here last divisor is 12

Hence,

HCF(120, 144, 204) = 12

Answer 2

$\sf{\pink{\underline{\underline{\purple{SOLUTION:-}}}}}$

[1] H.C.F of the given numbers by Prime Factorisation Method :-

(i) 28 , 36

Step - 1 :- To find LCM of 28 and 36, write each number as a product of prime factors .

$\rm\begin{array}{r | 1} 2 & 28 \\ \cline{2 - 2} 2 & 14 \\ \cline{2 - 2} & 7 \end{array}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{r | 1} 2 & 36 \\ \cline{2 - 2} 2 & 18 \\ \cline{2 - 2} 3 & 9 \\ \cline{2 - 2} & 3 \end{array}$

☃️ 28 = 2 × 2 × 7 = 2² × 7

☃️ 36 = 2 × 2 × 3 × 3 = 2² × 3²

Step - 2 :- Multiply all the common prime factors with the lowest degree .

✍️ Here we have only 2 as a common prime factor with the lowest power of 2 .

$\rm\red{\therefore}$ H.C.F of 28 and 36 = 2² = 4

(ii) 54 , 72 , 90

$\rm\begin{array}{r | 1} 2 & 54 \\ \cline{2 - 2} 3 & 27 \\ \cline{2 - 2} 3 & 9 \\ \cline{2 - 2} & 3 \end{array}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{r | 1} 2 & 72 \\ \cline{2 - 2} 2 & 36 \\ \cline{2 - 2} 2 & 18 \\ \cline{2 - 2} 3 & 9 \\ \cline{2 - 2} & 3 \end{array}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{r | 1} 2 & 90 \\ \cline{2 - 2} 3 & 45 \\ \cline{2 - 2} 3 & 15 \\ \cline{2 - 2} & 5 \end{array}$

☃️ 54 = 2 × 3 × 3 × 3 = 2 × 3² × 3

☃️ 72 = 2 × 2 × 2 × 3 × 3 = 2 × 2² × 3²

☃️ 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5

$\rm\red{\therefore}$ H.C.F of 54 , 72 and 90 = 2 × 3² = 18

(iii) 105 , 140 , 175

$\rm\begin{array}{r | 1} 3 & 105 \\ \cline{2 - 2} 5 & 35 \\ \cline{2 - 2} & 7 \end{array}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{r | 1} 2 & 140 \\ \cline{2 - 2} 2 & 70 \\ \cline{2 - 2} 5 & 35 \\ \cline{2 - 2} & 7 \end{array}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{r | 1} 5 & 175 \\ \cline{2 - 2} 5 & 35 \\ \cline{2 - 2} & 7 \end{array}$

☃️ 105 = 3 × 5 × 7

☃️ 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7

☃️ 175 = 5 × 5 × 7

$\rm\red{\therefore}$ H.C.F of 105 , 140 and 175 = 5 × 7 = 35

_________________________________

[2] H.C.F of the given numbers by Division Method :-

Step - 1 :- Divide the largest number by the smallest number .

Step - 2 :- Take divisor as new dividend and remainder as the new divisor, i.e. divide the first divisor by the first remainder .

Step - 3 :- Proceed till the remainder is zero and the last divisor will be the HCF of the given numbers .

(i) 198 , 429

=> 429/198 :- Quotient = 2 , Reminder = 33

Again,

=> 198/33 :- Quotient = 6 , Reminder = 0

$\rm\red{\therefore}$ H.C.F of 198 and 429 = 33

(ii) 20 , 64 , 104

=> 64/20 :- Quotient = 3 , Reminder = 4

Again,

=> 20/4 :- Quotient = 5 , Reminder = 0

Now,

=> 104/4 :- Quotient = 26 , Reminder = 0

$\rm\red{\therefore}$ H.C.F of 20 , 64 and 104 = 4

(iii) 120 , 144 , 204

=> 144/120 :- Quotient = 1 , Reminder = 24

Again,

=> 120/24 :- Quotient = 5 , Reminder = 0

Now,

=> 204/24 :- Quotient = 8 , Reminder = 12

Again,

=> 24/12 = :- Quotient = 2 , Reminder = 0

$\rm\red{\therefore}$ H.C.F of 120 , 144 and 204 = 12

⭐ See the attachment picture for better understanding of Division Method .