Introduction:
To find the smallest positive whole number that is exactly divisible by every single-digit whole number except 5, we need to find the least common multiple (LCM) of the numbers 1, 2, 3, 4, 6, 7, 8, and 9.

Prime Factorization:
To find the LCM, we first need to find the prime factorization of each number. Prime factorization is the process of breaking down a number into its prime factors. Let's find the prime factorization of each number:

1 = 1 (no prime factors)
2 = 2
3 = 3
4 = 2 * 2
6 = 2 * 3
7 = 7
8 = 2 * 2 * 2
9 = 3 * 3

LCM Calculation:
To find the LCM, we need to consider the highest power of each prime factor. The LCM will be the product of these prime factors raised to their highest powers.

The prime factors we have are 2, 3, and 7. We need to consider the highest power of each prime factor:

Highest power of 2 = 2 * 2 * 2 = 8
Highest power of 3 = 3 * 3 = 9
Highest power of 7 = 7

Now, we can calculate the LCM by multiplying the highest powers of the prime factors:

LCM = 2 * 2 * 2 * 3 * 3 * 7 = 504

Conclusion:
Therefore, the smallest positive whole number that is exactly divisible by every single-digit whole number except 5 is 504.

2520