Question

Find the HCF and LCM of 90 and 144 by prime factorization method.

Answer 1

Hint: In order to solve this problem we need to use factor tree for prime factorization. Prime factorization is the factorization of the number in which all the factors are prime numbers. HCF is the highest common factor of a number and LCM is the lowest common factor of the number. Doing this will solve your problem and will give you the right answer.

Complete step-by-step answer:
HCF - The greatest number which divides each of the two or more numbers is called HCF or Highest Common Factor.
LCM - One way to find the least common multiple of two numbers is to first list the prime factors of each number. Then multiply each factor the greatest number of times it occurs in either number.
Using the factor tree for the prime factorization of 90 and 144.
We have,
90 = $2 \times {3^2} \times 5$
144 = ${2^4} \times {3^2}$
To find the HCF, we list the common prime factor and their smallest exponents in 90 and 144 as under:
Common prime factor = 2, 3
Least exponents that is the least amount of power we have = 1, 2
HCF = ${2^1} \times {3^2}$ = 9 x 2 = 18
To find LCM we have to make a list of all prime factors of 90 and 144, their greatest exponents as follows:
Prime factor of 90 and 144 = 2, 3, 5
Greatest exponents = 4, 2, 1
LCM = ${2^4} \times {3^2} \times {5^1}$
LCM = 720
So, the answer is HCF = 18 and LCM = 720.

Note: Whenever we face such types of problems we use some important points. Like that we always use the factor tree and first of all find the prime factor of numbers then write the greatest and smallest exponents of prime factor then find HCF using smallest exponents and LCM by greatest exponents of prime factor.