To calculate the LCM of 36 and 90 by the division method, we will divide the numbers(36, 90) by their prime factors (preferably common). The product of these divisors gives the LCM of 36 and 90.

To find the LCM of 36 and 90 using prime factorization, we will find the prime factors, (36 = 2 × 2 × 3 × 3) and (90 = 2 × 3 × 3 × 5). LCM of 36 and 90 is the product of prime factors raised to their respective highest exponent among the numbers 36 and 90. ⇒ LCM of 36, 90 = 22 × 32 × 51 = 180.

The LCM of 36 and 90 is 180. To find the LCM (least common multiple) of 36 and 90, we need to find the multiples of 36 and 90 (multiples of 36 = 36, 72, 108, 144 . . . . 180; multiples of 90 = 90, 180, 270, 360) and choose the smallest multiple that is exactly divisible by 36 and 90, i.e., 180.

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The LCM of two non-zero integers, x(36) and y(90), is the smallest positive integer m(180) that is divisible by both x(36) and y(90) without any remainder.

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The least number divisible by 36 and 90 = LCM(36, 90) LCM of 36 and 90 = 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 5] ⇒ Least perfect square divisible by each 36 and 90 = LCM(36, 90) × 5 = 900 [Square root of 900 = √900 = ±30] Therefore, 900 is the required number.

Let the other number be b. ∵ GCD × LCM = 90 × b ⇒ b = (GCD × LCM)/90 ⇒ b = (18 × 180)/90 ⇒ b = 36 Therefore, the other number is 36.

Given: GCD = 18 product of numbers = 3240 ∵ LCM × GCD = product of numbers ⇒ LCM = Product/GCD = 3240/18 Therefore, the LCM is 180. The probable combination for the given case is LCM(36, 90) = 180.

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LCM of 36 and 90 is the smallest number among all common multiples of 36 and 90. The first few multiples of 36 and 90 are (36, 72, 108, 144, . . . ) and (90, 180, 270, 360, . . . ) respectively. There are 3 commonly used methods to find LCM of 36 and 90 - by division method, by prime factorization, and by listing multiples.

The LCM of 36 and 90 is the product of all prime numbers on the left, i.e. LCM(36, 90) by division method = 2 × 2 × 3 × 3 × 5 = 180.

LCM(90, 36) × GCF(90, 36) = 90 × 36 Since the LCM of 90 and 36 = 180 ⇒ 180 × GCF(90, 36) = 3240 Therefore, the GCF (greatest common factor) = 3240/180 = 18.

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Prime factorization of 36 and 90 is (2 × 2 × 3 × 3) = 22 × 32 and (2 × 3 × 3 × 5) = 21 × 32 × 51 respectively. LCM of 36 and 90 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 32 × 51 = 180. Hence, the LCM of 36 and 90 by prime factorization is 180.