Question

Find the lcm and hcf of 336 and 54 and verify that lcm × hcf = product of the two numbers

Answer 1

Answer:

The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.

LCM × HCF = Product of the Numbers

Suppose A and B are two numbers, then.

LCM (A & B) × HCF (A & B) = A × B

Example 1: Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12

Solution: LCM and HCF of 9 and 12:

9 = 3 × 3 = 3²

12 = 2 × 2 × 3 = 2² × 3

LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 9 and 12 = 3

LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108

Product of 9 and 12 = 9 × 12 = 108

Hence, LCM (9 & 12) × HCF (9 & 12) = 108 = 9 × 12

Property 2: HCF of co-prime numbers is 1. Therefore LCM of given co-prime numbers is equal to the product of the numbers.

LCM of Co-prime Numbers = Product Of The Numbers

Example 2: 8 and 9 are two co-prime numbers. Using this numbers verify, LCM of Co-prime Numbers = Product Of The Numbers

Solution: LCM and HCF of 8 and 9:

8 = 2 × 2 × 2 = 2³

9 = 3 × 3 = 3²

LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72

HCF of 8 and 9 = 1

Product of 8 and 9 = 8 × 9 = 72

Hence, LCM of co-prime numbers = Product of the numbers

Property 3: H.C.F. and L.C.M. of Fractions

LCM of fractions = LCMofnumeratorsHCFofdenominators

HCF of fractions = HCFofnumeratorsLCMofdenominators

Example 3: Find the HCF of 1225, 910, 1835, 2140

Solution: The required HCF is = HCFof12,9,18,21LCMof25,10,35,40 = 31400

To solve more problems on HCF and LCM download BYJU’S – The Learning App from Google Play Store and watch interactive videos. Also, take free tests to practice for exams.

Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

$\begin{array}{r | 1} 2 & 510 \\ \cline{2-2} 2 & 255 \\ \cline{2-2} 5 & 85 \\ \cline{2-2} & 17 \end{array}$

$\begin{array}{r | 1} 2 & 336 \\ \cline{2-2} 2 & 168 \\ \cline{2-2} 2 & 84 \\ \cline{2-2} 2 & 42 \\ \cline{2-2} 3 & 21 \\ \cline{2-2} & 7 \end{array}$

$\tt{336=2^4\times 3\times 7}$

$\tt{54=2\times 3^3}$

HCF(336,54) = 2 × 3 = 6

LCM(336,54)

$\tt{2^4\times 3^3\times 7}$

= 3024

★Verification

LCM × HCF = 3024 × 6 = 18144

336 × 54 = 18144

★Hence Proved

LCM × HCF = Product of two numbers.