Let's find the H.C.F. of the following numbers by factorisation method. e) 36, 48 f) 30, 75, 90 g) 32, 64, 80

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here are the solutions for questions 4 (continued) and 5:

4. Let's find the H.C.F. of the following numbers by factorisation method.

e) 36, 48

Find the prime factorization of each number.
$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$ $48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1$
Identify the common prime factors and their lowest powers.
The common prime factors are $2^2$ and $3^1$ .
Multiply these common prime factors.
$\text{H.C.F.} = 2^2 \times 3^1 = 4 \times 3 = 12$
H.C.F. of 36 and 48 is $\boxed{\text{12}}$ .

f) 30, 75, 90
Find the prime factorization of each number.
$30 = 2 \times 3 \times 5 = 2^1 \times 3^1 \times 5^1$ $75 = 3 \times 5 \times 5 = 3^1 \times 5^2$ $90 = 2 \times 3 \times 3 \times 5 = 2^1 \times 3^2 \times 5^1$
Identify the common prime factors and their lowest powers.
The common prime factors are $3^1$ and $5^1$ .
Multiply these common prime factors.
$\text{H.C.F.} = 3^1 \times 5^1 = 3 \times 5 = 15$
H.C.F. of 30, 75, and 90 is $\boxed{\text{15}}$ .

g) 32, 64, 80
Find the prime factorization of each number.
$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$ $64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6$ $80 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5^1$
Identify the common prime factors and their lowest powers.
The common prime factor is $2^4$ .
Calculate the value.
$\text{H.C.F.} = 2^4 = 16$
H.C.F. of 32, 64, and 80 is $\boxed{\text{16}}$ .

h) 96, 144, 216
Find the prime factorization of each number.
$96 = 2^5 \times 3^1$ $144 = 2^4 \times 3^2$ $216 = 2^3 \times 3^3$
Identify the common prime factors and their lowest powers.
The common prime factors are $2^3$ and $3^1$ .
Multiply these common prime factors.
$\text{H.C.F.} = 2^3 \times 3^1 = 8 \times 3 = 24$
H.C.F. of 96, 144, and 216 is $\boxed{\text{24}}$ .

5. Let's find the H.C.F. of the following numbers by division method.

a) 18, 24
Divide the larger number (24) by the smaller number (18).
$24 = 18 \times 1 + 6$
Divide the previous divisor (18) by the remainder (6).
$18 = 6 \times 3 + 0$
The last non-zero remainder is the H.C.F.
H.C.F. of 18 and 24 is $\boxed{\text{6}}$ .

b) 28, 32
Divide 32 by 28.
$32 = 28 \times 1 + 4$
Divide 28 by 4.
$28 = 4 \times 7 + 0$
H.C.F. of 28 and 32 is $\boxed{\text{4}}$ .

c) 36, 42
Divide 42 by 36.
$42 = 36 \times 1 + 6$
Divide 36 by 6.
$36 = 6 \times 6 + 0$
H.C.F. of 36 and 42 is $\boxed{\text{6}}$ .

d) 52, 65
Divide 65 by 52.
$65 = 52 \times 1 + 13$
Divide 52 by 13.
$52 = 13 \times 4 + 0$
H.C.F. of 52 and 65 is $\boxed{\text{13}}$ .

e) 35, 56, 70
Find the H.C.F.
of the first two numbers, 35 and 56.
Divide 56 by 35:
$56 = 35 \times 1 + 21$
Divide 35 by 21:
$35 = 21 \times 1 + 14$
Divide 21 by 14:
$21 = 14 \times 1 + 7$
Divide 14 by 7:
$14 = 7 \times 2 + 0$
The H.C.F. of 35 and 56 is 7.
Find the H.C.F.
of the result (7) and the third number (70).
Divide 70 by 7:
$70 = 7 \times 10 + 0$
H.C.F. of 35, 56, and 70 is $\boxed{\text{7}}$ .

f) 30, 45, 60
Find the H.C.F.
of the first two numbers, 30 and 45.
Divide 45 by 30:
$45 = 30 \times 1 + 15$
Divide 30 by 15:
$30 = 15 \times 2 + 0$
The H.C.F. of 30 and 45 is 15.
Find the H.C.F.
of the result (15) and the third number (60).
Divide 60 by 15:
$60 = 15 \times 4 + 0$
H.C.F. of 30, 45, and 60 is $\boxed{\text{15}}$ .

g) 156, 221, 390
Find the H.C.F.
of the first two numbers, 156 and 221.
Divide 221 by 156:
$221 = 156 \times 1 + 65$
Divide 156 by 65:
$156 = 65 \times 2 + 26$
Divide 65 by 26:
$65 = 26 \times 2 + 13$
Divide 26 by 13:
$26 = 13 \times 2 + 0$
The H.C.F. of 156 and 221 is 1

Let's find the H.C.F. of the following numbers by factorisation method. e) 36, 48 f) 30, 75, 90 g) 32, 64, 80

Step-by-step solution

Find the prime factorization of each number.

Identify the common prime factors and their lowest powers.

Multiply these common prime factors.

Find the prime factorization of each number.

Identify the common prime factors and their lowest powers.

Multiply these common prime factors.

Find the prime factorization of each number.

Identify the common prime factors and their lowest powers.

Calculate the value.

Find the prime factorization of each number.

Identify the common prime factors and their lowest powers.

Multiply these common prime factors.

Divide the larger number (24) by the smaller number (18).

Divide the previous divisor (18) by the remainder (6).

Divide 32 by 28.

Divide 28 by 4.

Divide 42 by 36.

Divide 36 by 6.

Divide 65 by 52.

Divide 52 by 13.

Find the H.C.F.

Find the H.C.F.

Find the H.C.F.

Find the H.C.F.

Find the H.C.F.

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