Problem:
Given that (232 - 1) is completely divisible by a whole number, we need to determine which of the following numbers is also completely divisible by this number.

Solution:

To solve this problem, we need to understand a few concepts related to divisibility.

Concept 1: Divisibility Rule for (a - b) with respect to c:
If (a - b) is completely divisible by c, then a and b leave the same remainder when divided by c.

Concept 2: Divisibility Rule for (a - 1) with respect to a prime number:
If (a - 1) is completely divisible by a prime number, then a is not divisible by that prime number.

Concept 3: Prime Factorization:
Every positive integer can be expressed as a product of prime numbers raised to some powers. This is known as prime factorization.

Now let's analyze each option to determine which number is completely divisible by (232 - 1).

Option a) (216 - 1):
To determine if (216 - 1) is completely divisible by (232 - 1), we can use Concept 1. If both numbers leave the same remainder when divided by (232 - 1), then (216 - 1) is completely divisible by (232 - 1).

Let's calculate the remainders:
216 % (232 - 1) = 216
1 % (232 - 1) = 1

Since the remainders are not the same, (216 - 1) is not completely divisible by (232 - 1). Therefore, option a) is not the correct answer.

Option b) (216 - 1):
We can use the same approach as above to determine if (216 - 1) is completely divisible by (232 - 1).

Let's calculate the remainders:
216 % (232 - 1) = 216
1 % (232 - 1) = 1

Again, the remainders are not the same, so (216 - 1) is not completely divisible by (232 - 1). Thus, option b) is not the correct answer.

Option c) (7 x 223):
To determine if (7 x 223) is completely divisible by (232 - 1), we can use Concept 2. If (232 - 1) is completely divisible by either 7 or 223, then (7 x 223) is not completely divisible by (232 - 1).

Let's check the divisibility of (232 - 1) by 7:
(232 - 1) % 7 = 6

Since the remainder is not 0, (7 x 223) is not completely divisible by (232 - 1). Hence, option c) is not the correct answer.

Option d) (296 - 1):
Let's apply the same approach to determine if (296 - 1) is completely divisible by (232 - 1).

Let's calculate the remainders:
296 % (232 - 1) = 5
1 % (232 - 1) = 1

The remainders are not the same, so (296 - 1) is not