Question

LCM of 420 & 272 by fundamental theorem of arithmetic,​

Answer 1

To Find :

• LCM of two numbers 420 and 272

Given :

• Two numbers are given .

420 can be expressed as :-

= 2 × 2 × 3 × 5 × 7

= 2² × 3 × 5 × 7

272 can be expressed as :-

= 2 × 2 × 2 × 2 × 17

= 2⁴ × 17

So,

• HFC of two numbers is 4

LCM of two numbers is 2 × 2 × 2 × 2 × 3 × 5 × 7 × 17

= 2⁴ × 3 × 5 × 7 × 17

= 16 × 15 × 119

= 28560

⋆ Verification :

we know that,

HCF × LCM = product of two numbers

• HCF = 4
• LCM = 28560
• Two numbers = 420 ,272

⇛4 × 28560 = 420 × 272

⇛114240 = 114240

LHS = RHS

Hence, verified .

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Fundamental Theorem of Arithmetic :

Every composite number can be uniquely expressed as a product of primes , except for the order in which these prime factors occurs.

Some Examples :-

12 can be expressed as = 2 × 2 × 3

612 = 2× 2 × 3 × 3 × 17 = 2² × 3² × 17

1314 = 2 × 3 × 3 × 73 = 2 × 3² × 73

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Answer 2

Attachment LCM of 420 & 272

$\mathcal{\huge{\fbox{\red{Correct\:Question\:?}}}}$

⭐ Find the LCM of 420 & 272 by fundamental theorem of arithmetic.

$\mathcal{\huge{\fbox{\green{Answer:-}}}}$

➡ The LCM (420,272) is 28560.

$\mathcal{\huge{\fbox{\purple{Solution:-}}}}$

LCM of 420 & 272 by fundamental theorem of arithmetic.

According to the question,

$\begin{array}{r | l}</p><p></p><p>2 & 420 \\</p><p></p><p>\cline{2-2} 2 & 210 \\</p><p></p><p>\cline{2-2} 5 & 105 \\</p><p></p><p>\cline{2-2} 3 & 21 \\</p><p></p><p>\cline{2-2} 7 & 7 \\</p><p></p><p>\cline{2-2} & 1</p><p></p><p>\end{array}$

$\begin{array}{r | l}</p><p></p><p>2 & 272 \\</p><p></p><p>\cline{2-2} 2 & 136 \\</p><p></p><p>\cline{2-2} 2 & 68 \\</p><p></p><p>\cline{2-2} 2 & 34 \\</p><p></p><p>\cline{2-2} 17 & 17 \\</p><p></p><p>\cline{2-2} & 1</p><p></p><p>\end{array}$

420 = 2 × 2 × 3 × 5 × 7. = 2² × 3¹ × 5¹ × 7¹

272 = 2 × 2 × 2 × 2 × 17 = (2)^4 × 17¹

L.C.M(420,272) = (2)^4 × 5¹ × 7¹ × 17¹

= 16 × 3 × 5 × 7 × 17

= 48 × 35 × 17

=1680 × 17

= 28560

♣ Fundamental Theorem Of Arithmetic :-

• It states that every composite number can be factorised as a product of primes and this factorization is unique apart from the order in which the prime factor occurs.

⏩It's also came to be known as unique factorization theorem.

⏩Composite number is the product of prime numbers.

⏩ If any Integer is < 1, it is either la prime number or a product of prime factors.

⭕ HCF of two or more numbers is the product of the smallest power of each common prime factor involve with in a numbers.

⭕ LCM of two or more numbers is the product of the greatest power of each prime factor involve within a numbers of high power.

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