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To calculate the LCM of 30 and 42 by the division method, we will divide the numbers(30, 42) by their prime factors (preferably common). The product of these divisors gives the LCM of 30 and 42.

LCM of 30 and 42 is the smallest number among all common multiples of 30 and 42. The first few multiples of 30 and 42 are (30, 60, 90, 120, . . . ) and (42, 84, 126, 168, 210, 252, 294, . . . ) respectively. There are 3 commonly used methods to find LCM of 30 and 42 - by division method, by prime factorization, and by listing multiples.

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The LCM of 30 and 42 is 210. To find the LCM (least common multiple) of 30 and 42, we need to find the multiples of 30 and 42 (multiples of 30 = 30, 60, 90, 120 . . . . 210; multiples of 42 = 42, 84, 126, 168 . . . . 210) and choose the smallest multiple that is exactly divisible by 30 and 42, i.e., 210.

The LCM of 30 and 42 is the product of all prime numbers on the left, i.e. LCM(30, 42) by division method = 2 × 3 × 5 × 7 = 210.

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The LCM of two non-zero integers, x(30) and y(42), is the smallest positive integer m(210) that is divisible by both x(30) and y(42) without any remainder.

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The relation between GCF and LCM of 30 and 42 is given as, LCM(30, 42) × GCF(30, 42) = Product of 30, 42 Prime factorization of 30 and 42 is given as, 30 = (2 × 3 × 5) = 21 × 31 × 51 and 42 = (2 × 3 × 7) = 21 × 31 × 71 LCM(30, 42) = 210 GCF(30, 42) = 6 LHS = LCM(30, 42) × GCF(30, 42) = 210 × 6 = 1260 RHS = Product of 30, 42 = 30 × 42 = 1260 ⇒ LHS = RHS = 1260 Hence, verified.

LCM(42, 30) × GCF(42, 30) = 42 × 30 Since the LCM of 42 and 30 = 210 ⇒ 210 × GCF(42, 30) = 1260 Therefore, the greatest common factor (GCF) = 1260/210 = 6.

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Let the other number be y. ∵ GCD × LCM = 30 × y ⇒ y = (GCD × LCM)/30 ⇒ y = (6 × 210)/30 ⇒ y = 42 Therefore, the other number is 42.

Given: GCD = 6 product of numbers = 1260 ∵ LCM × GCD = product of numbers ⇒ LCM = Product/GCD = 1260/6 Therefore, the LCM is 210. The probable combination for the given case is LCM(30, 42) = 210.

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Prime factorization of 30 and 42 is (2 × 3 × 5) = 21 × 31 × 51 and (2 × 3 × 7) = 21 × 31 × 71 respectively. LCM of 30 and 42 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 21 × 31 × 51 × 71 = 210. Hence, the LCM of 30 and 42 by prime factorization is 210.

To find the LCM of 30 and 42 using prime factorization, we will find the prime factors, (30 = 2 × 3 × 5) and (42 = 2 × 3 × 7). LCM of 30 and 42 is the product of prime factors raised to their respective highest exponent among the numbers 30 and 42. ⇒ LCM of 30, 42 = 21 × 31 × 51 × 71 = 210.

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