Find the LCM 6a^2 , 2

Explanation of the Question:

The question asks for the calculation of the Least Common Multiple (LCM) of two algebraic expressions, specifically 6a^2 and 2. The LCM is the smallest non-zero common multiple of both expressions, which in this case requires finding a value that can be divided by both expressions without leaving a remainder. The process generally involves identifying the highest powers of the variables and the factors involved in both terms to determine the least common multiple that has each factor at least as many times as either expression. This would involve algebraic manipulation and understanding of prime factorization for numerical coefficients.

$6 a^{2}$ , $2$

Answer

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Solution:

Step 1:

To determine the LCM of $6 a^{2}$ and $2$ , we must consider both the numerical and the algebraic components separately. First, we calculate the LCM of the numerical values $6$ and $2$ , and then we find the LCM for the algebraic part, which is $a^{2}$ .

Step 2:

The LCM is the smallest number into which each of the given numbers can be divided without a remainder. The process involves two main actions: (1) Identifying the prime factors of each number, (2) Multiplying the prime factors together, choosing the highest power of each prime factor present in any of the numbers.

Step 3:

The number $6$ can be factored into primes as $2 \times 3$ .

Step 4:

The number $2$ is already a prime number, with prime factors being $1$ and $2$ itself.

Step 5:

To find the LCM of $6$ and $2$ , we take the highest power of each prime factor from both numbers. In this case, it is $2 \times 3$ .

Step 6:

Calculating the product of these prime factors gives us $2 \times 3 = 6$ .

Step 7:

For the variable part $a^{2}$ , the factors are simply $a$ repeated twice, as $a^{2}$ is $a$ times $a$ .

Step 8:

The LCM of the variable part $a^{2}$ is just $a^{2}$ , as there are no other variable factors to consider.

Step 9:

By multiplying $a$ by itself, we affirm that $a^{2}$ is the LCM of the algebraic part.

Step 10:

Finally, the LCM of $6 a^{2}$ and $2$ is found by combining the LCM of the numerical part with the LCM of the variable part, resulting in $6 a^{2}$ .

Knowledge Notes:

The Least Common Multiple (LCM) of two or more algebraic expressions is the smallest expression that is a multiple of each of the expressions. To find the LCM:

Separate the problem into its numerical and variable parts.
For the numerical part, factor each number into its prime factors.
For the variable part, write down the variables with their exponents.
The LCM is then the product of the highest powers of prime factors and variables from the given expressions.

Prime Factorization: This is the process of breaking down a composite number into its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

LCM of Numbers: The LCM of two or more numbers is the smallest multiple that is exactly divisible by each of the numbers.

LCM of Algebraic Expressions: When dealing with algebraic expressions, the LCM includes both the numerical coefficients and the variables with their exponents. The LCM of variables is the variable raised to the highest power that occurs in any of the expressions.

Combining LCMs: After finding the LCMs of the numerical and variable parts separately, they are multiplied together to get the overall LCM of the algebraic expressions.