Question

Find the LCM and HCF of the following pairs of integers and verify that LCM ×HCF= product of the two number ?
by using factor - tree method.
1. 336 and 54.​

Answer 1

Question:-

Find the LCM and HCF of the following pairs of integers and verify that LCM ×HCF= product of the two number ?

by using factor - tree method.

1. 336 and 54.

Answer:

(iii) Given : First number : 336

Second number : 54

The prime factors of 336 and 54 are :

336 = 2 x 2 x 2 x 2 x 3 x 7 = 2⁴ × 3¹ × 7¹

54 = 2 x 3 x 3 x 3 = 2¹ × 3³

L.C.M (336 , 54) = 2⁴ × 3³× 7 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7

L.C.M( 336 , 54) = 3024

[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]

H.C.F (336 , 54) = 2 × 3

H.C.F (336 , 54) = 6

[HCF(Highest Common Factor) of two or more numbers = Product of the smallest Power of each common prime factor involved in the numbers.]

We know that, L.C.M x H.C.F = First Number x Second Number

3024 x 6 = 336 x 54

18144 = 18144

Hence verified

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Answer 2

$\huge\fbox \red{Solution:}$

H.C.F.

For Finding HCF of 336 and 54 , let us first do the prime factorization of 336 and 54

♢ 336 = 2 × 2 × 2 × 2 × 3 ×7

♢ 54 = 2 × 3 × 3 × 3

∴ The HCF of 336 and 54 = 2 × 3

$\red\bigstar \rm \blue{ The \: HCF \: of \: 336 \: and \: 54 \:  = 6}</p><p>$

L.C.M.

For also finding the LCM of 336 and 54 , let us first do the prime factorization of 336 and 54

✩ 336 = 2 × 2 × 2 × 2 × 3 × 7

✩ 54 = 2 × 3 × 3 × 3

∴ The LCM of 336 and 54 = 2 × 3 × 2 × 2 × 2 × 7 × 3 × 3

$\red\bigstar \rm \blue{The \: LCM \: of \: 336 \: and \: 54  = 3024}$

$\bigstar \rm \purple{Verification }$

We have to verify that :-  LCM× HCF = product of the two numbers.

⋆LCM = 3024

⋆HCF = 6

⋆The Two numbers are 336 and 54

L.H.S

HCF × LCM = 6 × 3024

HCF × LCM = 18144

R.H.S

Product of the two numbers = 336 × 54

Product of the two numbers = 18144

$\fbox \blue{LHS=RHS}$

∴ Verified that LCM× HCF = Product of the two numbers