Question

Find the HCF and LCM of the following pairs of integers and verify that LCM×HCF =product of the two numbers: 45 and 165

Answer 1

Answer :-

HCF & LCM = 495 & 15 respectively.

$\rule{200}{1}$

$\qquad{\underline{\large{\bold\pink{|| \: Given \: || }}}}$

• Two numbers = 45, 165

$\qquad{\underline{\large{\bold\blue{|| \: To \: find \: || }}}}$

• HCF & LCM and verify : LCM × HCF = Product of numbers.

$\qquad{\underline{\large{\bold\blue{|| Solution || }}}}$

Here, the numbers are 45 & 165

We will find LCM by prime factorization:-

$\qquad\begin{array} {r | l} 3 & 45, 165 \\\cline{2 - 2} 5 & 15, 55 \\\cline{2 - 2} 3 & 3, 11 \\\cline{2 - 2} 11 & 1,1 \end{array}$

$\therefore\sf LCM = 3 \times 5 \times 3 \times 11 \\\\\therefore\large\underline{\boxed{\frak\blue{ \quad LCM = 495 \quad }}}$

$\rule{200}{1}$

Now finding HCF by prime factorization:-

$\sf 45 = 3 \times 3 \times 5 \\\\\sf 165 = 3 \times 5 \times 11 \\\\\sf We \: know, HCF = Product \: of \: common \: factors \\\\\longrightarrow\sf HCF = 3 \times 5 \\\\\therefore\large\underline{\boxed{\frak\blue{\quad HCF = 15 \quad }}}$

Therefore,

LCM & HCF : 495 & 15 respectively.

$\rule{200}{1}$

$\qquad{\underline{\large{\bold\red{|| Verification || }}}}$

→ HCF × LCM = Product of numbers

→ 495 × 15 = 45 × 165

→ 7425 = 7425

$\qquad{\underline{\large{\bold\green{|| Hence \: verified! || }}}}$

Answer 2

Solution :-

Given 2 Numbers are 45 & 165.

Prime Factors of Both Numbers :-

→ 45 = 3 * 3 * 5 = 3² * 5

→ 165 = 3 * 5 * 11

So,

→ HCF = 3 * 5 = 15 .

→ LCM = 3² * 5 * 11 = 9 * 5 * 11 = 495.

Verification :-

→ LCM×HCF = product of the two numbers.

Taking LHS,

→ LCM × HCF

→ 495 * 15

→ 7425 ---------------- Equation (1).

Taking RHS,

→ product of the two numbers

→ 45 * 165

→ 7425 ---------------- Equation (2).

→ Equation (1) = Equation (2) .

→ LHS = RHS (Hence, Verified).

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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 ) .

☛ HCM of fractions = ( HCM of Numerators ) / ( LCF of Denominators ) .

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