Question

If the HCF and LCM of two positive integers a and b are x and y ,then x²y²/a²b²=

Answer 1

$\large{\underline{\underline{\mathfrak{Answer :}}}}$

• Value of x²y²/a²b² = xy/ab

Or

• Value x²y²/a²b² = 1

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$\underline{\underline{\mathfrak{Step-By-Step-Explanation :}}}$

Given :

• 1st Integer = a
• 2nd Integer = b
• H.C.F = x
• L.X.M = y

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To Find :

• Value of x²*y²/a²b²

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Solution :

Use formula :

$\large{\boxed{\sf{L.C.M \: \times \: H.C.F \: = \: Product \: of \: numbers}}} \\ \\ \implies {\sf{xy \: = \: ab}} \\ \\ \footnotesize{\underline{\sf{\dag \: \: \: \: \: \: \: Multiply \: abxy \: both \: sides \: \: \: \: \: \: \:}}} \\ \\ \implies {\sf{xy \: \times \: abxy \: = \: ab \: \times \: abxy}} \\ \\ \implies {\sf{x^2y^2 \: \times \: ab \: = \: a^2 b^2 \: \times \: xy}} \\ \\ \implies {\sf{\dfrac{x^2y^2}{a^2 b^2} \: = \: \dfrac{xy}{ab}}} \\ \\ \Large{\boxed{\sf{\dfrac{x^2 y^2}{a^2 b^2} \: = \: \dfrac{xy}{ab}}}}$

$\rule{200}{2}$

$\implies {\sf{xy \: = \: ab}} \\ \\ \footnotesize{\underline{\sf{\dag \: \: \: \: \: \: \: Square \: Both \: sides \: \: \: \: \: \: \:}}} \\ \\ \implies {\sf{(xy)^2 \: = \: (ab)^2}} \\ \\ \implies {\sf{x^2 y^2 \: = \: a^2 b^2}} \\ \\ \implies {\sf{\dfrac{x^2 y^2}{a^2 b^2} \: = \: 1}} \\ \\ \Large{\boxed{\sf{\dfrac{x^2 y^2}{a^2 b^2} \: = \: 1}}} \\ \\ \underline{\sf{\therefore \: Value \: \: of \: \: \dfrac{x^2y^2}{a^2b^2} \: \: is \: \: \large{\sf{\dfrac{xy}{ab} \: \: or \: \: 1}} \: }}$

Answer 2

Given :-

• HCF and LCM of two positive integers a and b are x and y .

To Find :-

• (x²y²)/(a²b²) = ?

Concept Used :-

• The product of the H.C.F. and the L.C.M. of two numbers is equal to the product of the given numbers.

Solution :-

From Given Data we Have :-

→ First Number = a

→ Second Number = b

→ HCF of a & b = x

→ LCM of a & b = y .

So, From Above Told concept we can say that ,

→ First Number * Second Number = HCF * LCM .

Or,

→ ( a × b) = (x × y)

Squaring Both sides we get,

→ ( a × b)² = (x × y)²

→ a²b² = x²y² -------------- Equation (1)

Now, Putting value of Equation (1) we get,

→ (x²y²)/(a²b²)

→ (x²y²) / (x²y²)

→ 1 (Ans).

Or,

→ (a²b²) / (a²b²)

→ 1 (Ans).

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★★Extra Brainly Knowledge★★

✯✯ Some Properties of HCF & LCM ✯✯

☛ HCF(Highest Common Factor) :- The largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor).

☛ LCM(Least Common Multiple) :- The least or smallest common multiple of any two or more given natural numbers are termed as LCM.

☛ The H.C.F. of given numbers is not greater than any of the numbers.

☛ The L.C.M. of given numbers is not less than any of the given numbers.

☛ The H.C.F. of two co-prime numbers is 1.

☛ The L.C.M. of two or more co-prime numbers is equal to their product.

☛ If a number, say x, is a factor of another number, say y, then the H.C.F. of x and y is x and their L.C.M. is y.

☛ The product of the H.C.F. and the L.C.M. of two numbers is equal to the product of the given numbers. That is, if a and b are two numbers, then a x b = H.C.F. x L.C.M.

☛ LCM of fractions = ( LCM of Numerators ) / ( HCF of Denominators ) .

☛ HCF of fractions = ( HCF of Numerators ) / ( LCM of Denominators ) .

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