Question

If two positive integers a and b are written as a=x³y² and b = xy³ where x , y are prime numbers then find hcf a,b number

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Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept}}}$

Here the concept of HCF has been used. HCF is the highest common factor which means the largest common number which can completely divide the given set of numbers. We see that we are given prime factorisation of the numbers that means we have to find the common terms with their highest degree in prime factorisation of both which can divide the both terms.

Let's do it !!

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★ Solution :-

Given,

» a = x³ y ²

» b = x y³

We shall solve this question step by step.

• Step : 1

Now we see we can simplify the given terms more.

→ a = x × x × x × y × y

→ b = x × y × y × y

• Step : 2

This step includes the grouping of factors.

• For x ::

We see that in prime factorisation of a, there are three x terms and in prime factorisation of b, there is only one x term. So the highest degree of x which we can take as HCF will be x¹. This is because x¹ is common in both prime factorisation.

This means x¹ is one of the HCF of a and b.

• For y ::

Now we see that in prime factorisation of a, there are two y terms and in factorisation of b, there are three y terms. So the highest degree of y which we can take as HCF will be y². The reason because y² is common in both prime factorisation.

This means y² is one of the HCF of a and b.

• Step : 3

This step includes combining the HCF we got. This will give us,

✒ HCF(a, b) = x¹ × y²

✒ HCF(a, b) = x¹ y²

$\\\;\underline{\boxed{\tt{HCF(a,\:b)\;=\;\bf{\purple{x^{1}\:y^{2}}}}}}$

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★ Verification ::

For verification of our answer, we need to divide the terms by the HCf we got . Then,

• For a :-

$\\\;\tt{\Longrightarrow\;\;\dfrac{a}{HCF}}$

$\\\;\tt{\Longrightarrow\;\;\dfrac{x^{3}\:y^{2}}{x^{1}\:y^{2}}}$

Cancelling the terms, we get

$\\\;\tt{\Longrightarrow\;\;\orange{x^{2}}}$

• For b :-

$\\\;\tt{\Longrightarrow\;\;\dfrac{b}{HCF}}$

$\\\;\tt{\Longrightarrow\;\;\dfrac{x\:y^{3}}{x^{1}\:y^{2}}}$

Cancelling the terms, we get

$\\\;\tt{\Longrightarrow\;\;\red{y}}$

This proves the correctness of our answer. So our answer is correct.

✒ HCF(a, b) = x¹ y²

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★ More to know :-

$\\\;\sf{\leadsto\;\:LCM\:\times\:HCF\;=\;Product\;of\;Two\;Numbers}$

• LCM : It is the Lowest Common Multiple that is the smallest common number attained by multiplying two numbers with some other numbers.

Answer 2

$\huge \purple {\mathfrak{ \underline{ \underline{question: - }}}}$

If two positive integers a and b are written as a=x³y² and b = xy³ where x , y are prime numbers then find hcf a,b number.

$\huge \purple {\mathfrak{ \underline{ \underline{ \: required \: answer: - }}}}$

xy^2

$\huge \purple {\mathfrak{ \underline{ \underline{to \: find: - }}}}$

HCF (hєíght cσmmσn fαctσr ):- (αlѕσ cαllєd αѕ GCD) HCF of two numbers is the number which completely divides (with 0 remainder)

Bσth thє numвєrѕ.fσr єхαmplє,HCF of 14 and 21 is 7, HCF (54,27) is 27.

Given :-

a = x^3 y^2,b = xy^3

It cαn вє σвѕєrvєd thαt хч^2 íѕ α cσmmσn fαctσr σf вσth α αnd в.

Hєncє HCF (a,b) = xy^2.