Explanation:

Answer: (a)
The number has 7 digits, and has been denoted by: ABCDEFG
These letters can be replaced by 1, 2, 4, 5, 7, 8, 9, not necessarily in the same order.
We have to find the possible value of C + D + E
The original number (ABCDEFG) is divisible by 9. It has to be as 1 + 2 + 4 + 5 + 7 + 8 + 9 = 36, which is divisible by 9. This information is utterly useless.
After deleting 1 digit from the right, the resulting number (ABCDEF) is divisible by 6. It means that, F = 2, 4 or 8 (i.e. an even number).
Also, if even after removing G, the remaining number is divisible by 3, then it means G = 9.
After deleting 3 digits from the right, the resulting number (ABCD) is divisible by 4. It means that, D = 2, 4 or 8 (i.e. an even number).
After deleting 5 digits from the right, the resulting number (AB) is divisible by 2. It means that, B = 2, 4 or 8 (i.e. an even number).
So, F, D and B are even numbers (2, 4 or 8). And, A, C, E, and G are odd numbers (1, 5, 7 or 9).
After deleting 2 digits from the right, the resulting number (ABCDE) is divisible by 5. It means that, E = 5.
So, we just have to find C + D + E = C + D + 5, which must be an even number as C is odd (1, or 7), and D is even (2, 4, or 8).
On observing the options, we can see that C must be 1 and D must be 2. So, C + D + E = 1 + 2 + 5 = 8.
Note: This information was also redundant - After deleting 4 digits from the right, the resulting number (ABC) is divisible by 3. But it may be useful if we want to know the entire number. The only odd number remaining is 7. So, A = 7. So, the ABC is actually 7B1. B can be 4 or 8. But for 7B1 to be divisible by 3, B must be 4. So, F = 8. So, the seven-digit number is 7412589.