 # Find the LCM and HCF of the following integers by applying the prime factorization method. (1)12, 15 and 21 (2)17, 23 and 29 (3)8, 9 and 25

The full form of HCF is the Highest Common Factor.

The largest common divisor or gcd of two or more positive integers, as the mathematics rules dictate, appears to be the largest positive integer that divides the numbers without leaving a remainder.

Example:

Consider the two number 8 and 12.

As the highest number that can divide both 8 and 12 is 4.

Therefore, the H.C.F. of 8 and 12 would be 4.

The full form of LCM is Least common multiple.

In arithmetic, LCM (a,b) is the least common multiple of two numbers, a and b. And the LCM is the smallest or least positive integer that is divisible by both a and b.

Example:

Consider the two number 8 and 12 and let us write the multiples of two numbers.

Multiples of 8 = 16, 24, 32, 40 ,48, 56……

Multiples of 12 = 24, 36, 48, 60, 72………

From the above list, we can observe that least common multiples of 8 and 12 is 24.

I.e LCM (8, 12) = 24

(1) 12, 15 and 21

The prime factors of the given numbers are

12 = 2 x 2 x 3 = 2² × 3¹

15 = 3 x 5 =3¹ × 5¹

21 = 3 x 7 =3¹ × 7¹

HCF of 12, 15 and 21

HCF of (12, 15 & 21) = 3

LCM of 12, 15 and 21

LCM of (12, 15 & 21) = 2² × 2¹ × 5¹ × 3¹ × 7¹

LCM of (12, 15 & 21) = 420

(2) 17, 23 and 29

The prime factors of the given numbers are

17 = 1 x 17

23 = 1 x 23

29 = 1 x 29

HCF of 17, 23 and 29

HCF of (17, 23 & 29) = 1

LCM of 17, 23 and 29

LCM of 17, 23 and 29 = 23 x 29 x 1 x 17

LCM of (17, 23 & 29) = 11339

(3) 8, 9 and 25

The prime factors of the given numbers are

8 = 2 x 2 x2

8 = 2³

9 = 3 x 3

9 = 3²

25 = 5 x 5

25 = 5²

HCF of 8, 9 and 25

HCF of (8, 9 & 25) = 1

LCM of 8, 9 and 25

LCM of (8, 9 & 25) = 2 x 2 x2 × 3 x 3 × 5 x 5

LCM of (8, 9 & 25) = 1800.