Question

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is ____ .

Answer 1

Correct Answer: 6

To determine the number of natural numbers between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (without repetition) and divisible by 55, we consider the conditions for divisibility by 55. A number must be divisible by both 5 and 11.

1. For divisibility by 5, a number must end in 5.
2. To check divisibility by 11, alternate digits' sum difference must be a multiple of 11. 

We thus need to consider numbers of varying lengths formed with the digits 2, 3, 4, 5, 6:

Steps:

1. Four-digit numbers:
   • We use all five digits, forming a 5-digit number but ensuring they fall within our specified range.
   • End the number with 5 (since divisibility by 5 is required).
   • Possible numbers: 2345, 2365, 2453, 2456, 2463, 2563, 3452, and so on.
   • Utilize divisibility by 11 for verification.
2. Analysis:
   • Total combinations of 4 different digits using any 4 from the list except 5 in the end position.
   • Permutate remaining digits and check boundaries (1012 and 23421).

Ultimately, confirm the numbers adhere to 55: divisibility.

Let’s explicitly calculate and verify few options:

1. Using 2345: 2-4+5=3; hence, fails divisibility by 11.
2. Using 6455 is not valid as 5 must conclude.

Applying thorough checks on permutations:

Number	Divisible by 55
2453	No
2563	No
3452	Yes

Continue checking numbers falling into criteria.

After concise checks, numbers like 4562 validated fit conditions.

Upon extensive testing, precisely one entry meets all:

1. 3452 [Only fitting witihin]: 3+4-5+2=4.

Thus, the precise number of valid, natural digits within range that are divisible by 55 is:1

Answer 2

The correct answer is 6
Case I: When number is 4-digit number \((\overline{a\ b\ c\ d})\)
Here d is fixed and d=5
So, (a, b, c) can be (6, 4, 3), (3, 4, 6), (2, 3, 6), (6, 3, 2), (3, 2, 4) or (4, 2, 3)
⇒ 6 numbers
Case II: No number possible