(i)

The LCM must contain at least everything that is contained in the HCF.

Now, 2^5 already contains 2^4, so k does not need to contain any more 2. It may, or it may not, but to keep k as small as possible, k should not have any 2.

Similarly, k does not need to contain any 3 or 5. Since the LCM does not contain 7 and 23 yet, these must be included. So the minimum value of k is 7 x 23 = 161.

(ii) To find the LCM of the three numbers, we must take the highest powers of each prime factor and then multiply them together.

The LCM of the first two numbers is
2^5 x 3^3 x 5 x k.

The third number is 3^5 x 17 x 19.

The highest power os 2 is 5.
The highest power of 3 is 5.
5, 7, 17, 19 and 23 all have highest powers of 1 each.

So the LCM is 2^5 x 3^5 x 5 x 7 x 17 x 19 x 23.