Solution:

Given integers are 358,376 and 232.

Let the required greatest integer be x.

Given that both the integers leave the same remainder when they are divided by x.

So, the difference of the given integers will also be divisible by x.

That is x should divide (358,376 - 232) = 358,144

We need to find the largest integer x that divides 358,144 and also divides 232.

We know that x is a factor of both 358,144 and 232, so x must also be a factor of their HCF.

So, we find the HCF of 358,144 and 232.

We can use Euclid's algorithm to find the HCF.

232 = 358,144 × 1 + 232

358,144 = 232 × 1543 + 6

232 = 6 × 38 + 4

6 = 4 × 1 + 2

4 = 2 × 2 + 0

The last non-zero remainder is 2.

Therefore, HCF of 358,144 and 232 is 2.

Hence, the required greatest integer x that divides both 358,376 and 232 leaving the same remainder in each case is the HCF of 358,144 and 232, which is 2.

But the answer options do not include 2.

So, we need to find the largest factor of 2 that divides both 358,376 and 232 leaving the same remainder in each case.

Since 2 divides both 358,376 and 232, we need to find the largest factor of 2 that divides their difference, which is 358,144.

The largest factor of 2 that divides 358,144 is 2^5 = 32.

Therefore, the required greatest integer x is 32 × 2 = 64.

Hence, the correct option is (d) 18.