Question

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

Answer 1

Hint: Prime factorization method means splitting the given number in terms of prime factors. We know that the number which exactly divides two or more numbers is called the common factor. Find the common factor between two or more given numbers that is called HCF or highest common factor. From all the prime factors of the given number, identify the maximum number of times each prime factor appears. This is called LCM or least common multiple.

Complete step-by-step answer:
Using prime factorization method-
(1) 12, 15, 21
Factor of \[12=2\times 2\times 3\]
Factor of \[15=3\times 5\]
Factor of \[21=3\times 7\]
\[HCF\left( 12,15,21 \right)=3\]. . . . . . . . (1)
\[LCM\left( 12,15,21 \right)=2\times 2\times 3\times 5\times 7=420\]. . . . . . . (2)

(2) 17, 23, 29
Factor of \[17=1\times 17\]
Factor of \[23=1\times 23\]
Factor of \[29=1\times 29\]
\[HCF\left( 17,23,29 \right)=1\]. . . . . . . . (3)
\[LCM\left( 17,23,29 \right)=1\times 17\times 23\times 29=11339\]. . . . . . . (4)

(3) 8, 9, 25
Factor of \[8=2\times 2\times 2\times 1\]
Factor of \[9=3\times 3\times 1\]
Factor of \[25=5\times 5\times 1\]
\[HCF\left( 8,9,25 \right)=1\]. . . . . . . . . (5)
\[LCM\left( 8,9,25 \right)=2\times 2\times 2\times 3\times 3\times 5\times 5=1800\]. . . . . . . . . (6)

Note: HCF can also be found using the division method. In this method divide the largest number by the smallest number of the given numbers until the remainder is zero. The last divisor will be HCF of given numbers. LCM can also be found using divide the given numbers by using least common prime factor and continue till there is no more prime factor. Then multiply all the divisors.