Find the LCM of the numbers:

$$2^3 imes 9^2 imes 13, \quad 2^2 imes 13^2 imes 19, \quad 9^3 imes 13^2 imes 19^2$$

Question

Find the LCM of the numbers:

$$2^3 	imes 9^2 	imes 13, \quad 2^2 	imes 13^2 	imes 19, \quad 9^3 	imes 13^2 	imes 19^2$$

Accepted Answer

**Solution:**

1. Express each number in terms of prime factors:

- First number: $$2^3 	imes 9^2 	imes 13 = 2^3 	imes (3^2)^2 	imes 13 = 2^3 	imes 3^4 	imes 13^1$$

- Second number: $$2^2 	imes 13^2 	imes 19 = 2^2 	imes 13^2 	imes 19^1$$

- Third number: $$9^3 	imes 13^2 	imes 19^2 = (3^2)^3 	imes 13^2 	imes 19^2 = 3^6 	imes 13^2 	imes 19^2$$

2. To find the LCM, take the highest power of each prime factor appearing in any of the numbers:

- For prime 2: highest power is $$\max(3, 2, 0) = 3$$

- For prime 3: highest power is $$\max(4, 0, 6) = 6$$

- For prime 13: highest power is $$\max(1, 2, 2) = 2$$

- For prime 19: highest power is $$\max(0, 1, 2) = 2$$

3. Therefore, the LCM is:

$$2^3 	imes 3^6 	imes 13^2 	imes 19^2$$

4. Since $$9 = 3^2$$, we can write $$3^6 = (3^2)^3 = 9^3$$, so the LCM can also be expressed as:

$$2^3 	imes 9^3 	imes 13^2 	imes 19^2$$

**Answer:** $$2^3 	imes 9^3 	imes 13^2 	imes 19^2$$ (Option 1)

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