A nine digit number having all its digits distinct and none being zero can be partitioned into two as per well. It turns out that every time the left part is always divisible by the number of digits it has. Which of the following number satisfies this property?

a. 183654927
b. 381654729
c. 361854729
d. None of these

well, did you try any of them?

for (a)
1 divides 1
2 divides 18
3 divides 183
4 divides 1836
5 divides 18365
6 divides 183654
7 does not divide 1836549

so, try the others

A

Well, let's break it down, shall we?

Option a. 183654927:
If we divide the left part (183) by the total number of digits (9), we get 183/9 = 20.333... which is not an integer. So, option a is not divisible by its number of digits.

Option b. 381654729:
If we divide the left part (381) by the total number of digits (9), we get 381/9 = 42.333... which is also not an integer. So, option b is out as well.

Option c. 361854729:
If we divide the left part (361) by the total number of digits (9), we get 361/9 = 40.111... which is still not an integer. So, option c is not the one we're looking for.

Well, in this case, it seems like the answer is d. None of these options satisfy the given property. Maybe these numbers need a lesson in being divisible!

Remember, folks, numbers can be tricky, but having a sense of humor always helps!

To determine which number satisfies the given property, we need to check if the left part of the number is divisible by the number of digits it has.

Let's evaluate each option:

a. 183654927:
The left part is 183, which has 3 digits. However, 183 is not divisible by 3. So this number does not satisfy the property.

b. 381654729:
The left part is 381, which also has 3 digits. However, 381 is not divisible by 3. Thus, this number does not satisfy the property.

c. 361854729:
The left part is 361, which has 3 digits. Interestingly, 361 is divisible by 3 because 361 รท 3 = 120. This number satisfies the given property.

Therefore, the correct answer is option c. 361854729.

To solve this problem, we need to check whether the left part of the nine-digit number is always divisible by the number of digits it has.

Let's start by analyzing the numbers given as options:

a. 183654927
b. 381654729
c. 361854729

To check if a number is divisible by the number of its digits, we need to determine the number of its digits and then divide the left part of the number by that count.

For option a (183654927), the left part is 183, which has 3 digits. If we divide 183 by 3, we get 61. This means that the left part (183) is divisible by the number of digits it has (3).

For option b (381654729), the left part is 381, which has 3 digits. If we divide 381 by 3, we get 127. This means that the left part (381) is divisible by the number of digits it has (3).

For option c (361854729), the left part is 361, which also has 3 digits. If we divide 361 by 3, we get 120.33 (approximately). Since the result is not a whole number, the left part (361) is not divisible by the number of digits it has (3).

Based on this analysis, the only option that satisfies the given property is option b (381654729). Therefore, the correct answer is b. 381654729.