Question

My question is about LCM (Least Common Multiple)

So the thing is in this sum, in the first pic as u can see I took 3^2 as it is the least one from the two 3s. But, in the other sum, in the 2nd pic if i take 2 and then when I multiply by 2 the answer turns out wrong but If i multiply it with 2^2 and 3, the answer comes right. But my question is that why is it that I'm taking 2^2 but why not 2? 2 is the least one right and there is already 2^2 so why am I taking 2^2?

Answer 1

• 18a²b²c²
• 12x³y²z²

Explanation:

You want the least common multiple (LCM) of {6a²b, 9b²c, 3c²a} and of {4x²y²z², 6x³yz, 12xy²z}.

Least common multiple

The LCM of two or more terms is a product that has includes all of the factors of each of the terms. One way to find the LCM is to list the factors of the terms, identify the different ones and the powers of each, then choose the highest power of each of the different factors.

{6a²b, 9b²c, 3c²a}

6a²b — factors are ...

• 2
• 3
• a²
• b

9b²c — factors are ...

• 3²
• b²
• c

3c²a — factors are ...

• 3
• a
• c²

The distinct factors are ...

• 2 — power of 1; 1 is the highest power
• 3 — powers of 1 and 2; 2 is the highest power
• a — powers of 1 and 2; 2 is the highest power
• b — powers of 1 and 2; 2 is the highest power
• c — powers of 1 and 2; 2 is the highest power

The product of highest powers is the LCM:

LCM = 2¹·3²·a²·b²·c² = 18a²b²c²

{4x²y²z², 6x³yz, 12xy²z}

The distinct factors are ...

• 2 — powers of 1 and 2; 2 is the highest power
• 3 — power of 1; 1 is the highest power
• x — powers of 1, 2, and 3; 3 is the highest power
• y — powers of 1 and 2; 2 is the highest power
• z — powers of 1 and 2; 2 is the highest power

The product of the highest powers is ...

LCM = 2²·3¹·x³·y²·z² = 12x³y²z²

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Additional comment

Sometimes the process of finding the LCM can be confused with finding the GCF (greatest common factor). The GCF will be the product of the lowest of the powers of the distinct factors. That power may be zero for any given factor in any given term.

For example, in the first set of terms, the factor 2 has a power of 0 in the last two terms. The factor 'a' has a power of zero in the second term; b has a power of 0 in the third term; c has a power of 0 in the first term. Then the GCF is ...

2⁰·3¹·a⁰·b⁰·c⁰ = 3 . . . . . GCF of {6a²b, 9b²c, 3c²a}

In the second set of terms, 2 has a power of 1 in the 2nd term; 3 has a power of 0 in the 1st term; x, y, z all have powers of 1 in at least one of the terms. Then the GCF is ...

2¹·3⁰·x¹·y¹·z¹ = 2xyz . . . . . GCF of {4x²y²z², 6x³yz, 12xy²z}

In short, every term must be a factor of the LCM, and the GCF must be a factor of every term.

One way to find the LCM of two terms is to divide their product by their GCF. The process can be repeated to include more terms in the LCM.