Question: How many 8 digit numbers that are divisible by 9 can be formed so that no digit is repeated?

1. 3 (7!)
2. 4 (7!)
3. 8!
4. 36 (7!)
5. 9!

Answer:
Approach Solution (1):
Sum of all digits from 0 to 9 = 45 (divisibly by 9)
Two digits to be excluded out of 10 digits can be {0,9} or {1,8} or {2,7} or {3,6} or {4,5}
If {0,9} are not used, Total Numbers = 8!
If {1,8} are not used, Total Numbers = 7*7!
If {2,7} are not used, Total Numbers = 7*7!
If {3,6} are not used, Total Numbers = 7*7!
If {4,5} are not used, Total Numbers = 7*7!
Total numbers = 7*7!*4 + 8! = 36 * 7!

Correct option: D

Approach Solution (2):
Number of 8 digits in which no digit is repeated = 9 * 9 * 8 * 7 * 6 * 5 * 4 * 3
As sum of all 10 digits is multiple of 9 and sum of digits of required 8-digit numbers is also divisible by 9, sum of 2 excluded digits must be equal to 9
Total number of ways to select 2 distinct digits out of 10 = 10C2 = 45
Total number of ways to select 2 distinct digits out of 10 = 10C2 = 45
Total number of ways to select 2 distinct digits out of 10 such that their sum is 9 = 5
{(0,9), (1,8), (2,7), (3,6) and (4,5)}
Number of 8 digit numbers that are divisible by 9 can be formed so that no digit is repeated =
\(\frac{5}{45}\)9 *4 * 2 * 7 * 6 * 5 * 4 * 3
36 * 7!

Correct option: D

Approach Solution (3):
As (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 45 and 45 mod 9 = 0
Sum of 2 digits to be dropped (8 digit numbers to be formed) must be a multiple of 9 (obviously 9).
We get = 0 + 9 = 1 + 8 = 2 + 7 = 3 + 6 = 4 + 5
So, there are 5 cases
Now, number formed by removing (0,9) can be arranged in 8! Ways
But for other 4 cases, we have to exclude numbers having 0 as 8th digit (leading digit)
So, there are 8! – 7! = 7 (7!) ways for each case
We can form 8! + 28 (7!) = 36 (7!) numbers in this way

Correct option: D

“How many 8 digit numbers that are divisible by 9 can be formed so that no digit is repeated?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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