Find the hcf and lcm of 90 and 144 by the method of prime factorisation ?

Question

Answer

90= 2*3*3*5
144= 2*2*2*2*3*3

HCF= 2*3*3=18
LCM = 2^4 *3^2*5= 720

Answer

We have,
90= 2×3×3×5
144= 2×2×2×2×3×3
(H.C.F 90×144)=2×3×3 = 18
(L.C.M 90×144)= Product of give numbers/(H.C.F 90×144)
=90×144/18
=10×72
=720.

Answer

Prime Factorisation of 90 and 144:

To find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of 90 and 144, we need to first find the prime factorisation of both numbers.

Prime factorisation of 90:
To find the prime factorisation of 90, we divide it by the smallest prime number, which is 2. If 2 is a factor, we continue dividing by 2 until it is no longer a factor. Then, we move on to the next prime number, which is 3, and repeat the process.

90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5

Hence, the prime factorisation of 90 is 2 × 3 × 3 × 5, or written in exponent form, 2^1 × 3^2 × 5^1.

Prime factorisation of 144:
To find the prime factorisation of 144, we follow the same process.

144 ÷ 2 = 72
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3

Hence, the prime factorisation of 144 is 2 × 2 × 2 × 2 × 3 × 3, or written in exponent form, 2^4 × 3^2.

HCF (Highest Common Factor):

The HCF is the largest factor that is common to both numbers.

To find the HCF using prime factorisation, we need to identify the common prime factors and take their lowest exponent.

Common prime factors:
The common prime factors in the prime factorisation of 90 and 144 are 2 and 3.

Lowest exponent for 2:
The exponent for 2 in the prime factorisation of 90 is 1, and in the prime factorisation of 144, it is 4. So, we take the lowest exponent, which is 1.

Lowest exponent for 3:
The exponent for 3 in the prime factorisation of 90 is 2, and in the prime factorisation of 144, it is 2. So, we take the lowest exponent, which is 2.

Therefore, the HCF of 90 and 144 is 2^1 × 3^2, which is equal to 2 × 3 × 3 = 18.

LCM (Lowest Common Multiple):

The LCM is the smallest multiple that is common to both numbers.

To find the LCM using prime factorisation, we need to identify all the prime factors and take their highest exponent.

All prime factors:
The prime factors in the prime factorisation of 90 are 2, 3, and 5.
The prime factors in the prime factorisation of 144 are 2 and 3.

Highest exponent for 2:
The exponent for 2 in the prime factorisation of 90 is 1, and in the prime factorisation of 144, it

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