without actually calculating the LCM determine whether the LCM of 12 18 and 45 is divisible by 36

Answer: Yes, the LCM of $12$ , $18$ , and $45$ is divisible by $36$ .

Step-by-step solution

Step 1: Find the prime factorisation of the given numbers

We first express each of the given numbers, $12$ , $18$ , and $45$ , as a product of powers of primes using the prime factorisation method.

Step 2: Express the prime factorisation of 36

To check divisibility by $36$ , we find its prime factorisation. For any integer to be divisible by $36$ , its prime factorisation must contain at least $2^2$ and $3^2$ .

Step 3: Determine the prime factors in the LCM

By definition, the LCM is the product of the greatest power of each prime factor involved in the numbers. From $12$ , $18$ , and $45$ , the highest power of $2$ is $2^2$ , the highest power of $3$ is $3^2$ , and the highest power of $5$ is $5^1$ .

Step 4: Conclude divisibility using the Fundamental Theorem of Arithmetic

The prime factorisation of the LCM contains $2^2$ and $3^2$ , which means it contains $2^2 \times 3^2 = 36$ as a factor. Therefore, by the uniqueness guaranteed by the Fundamental Theorem of Arithmetic, the LCM of $12$ , $18$ , and $45$ is divisible by $36$ .

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