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If \[x = {2^3} \times 3 \times {5^2}\], \[y = {2^2} \times {3^3}\] , then HCF \[\left( {x,y} \right)\] is?

Last updated date: 20th Jun 2024
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Hint: Here in this question, we have to find the HCF (Highest common factor of x and y). first list out the common factors of both numbers x and y. Then take out the highest factor in the common factors of both numbers. That taken number will be a required HCF of a given two numbers x and y.

Complete step-by-step solution:
The largest positive integer which divides two or more integers without any remainder is called Highest Common Factor (HCF) or Greatest Common Divisor or Greatest Common Factor (GCF). To find HCF, we have two important methods which are the Prime factorisation method and division method.
We can also write the formula of HCF in terms of LCM are
\[HCF = \dfrac{{product}}{{LCM}}\]
Consider the given two numbers x and y
\[x = {2^3} \times 3 \times {5^2}\] and \[y = {2^2} \times {3^3}\]
Expand each prime factor of both numbers then
\[ \Rightarrow \,\,\,x = 2 \times 2 \times 2 \times 3 \times 5 \times 5\]
\[ \Rightarrow \,\,\,y = 2 \times 2 \times 3 \times 3 \times 3\]
Observing the factors of two numbers x and y
X having three 2 factor, one 3 factor and two 5 factor and
Y having two 2 factor and three 3 factors
In both x and y two 2 factor and one 3 factor had common, then common factors of x and y are
Common factors of \[\left( {x,y} \right) = 2 \times 2 \times 3\]
The Highest common factor, HCF of \[\left( {x,y} \right) = 12\].

Note: We must know about the multiplication, division and tables of multiplication to solve the question. We should divide by the number by the least number and hence it is the correct way to solve the problem. The LCM is abbreviated as Least common factor and the HCF is abbreviated as Highest common factor.