The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition not allowed) and divisible by 55 is \_\_\_\_\_\_\_.
Answer: 6
Solutions · 1
Official solution
For a 4-digit number abcd (using distinct digits from 2,3,4,5,6) divisible by 55, it must be divisible by 5 and 11. Since no digit is 5 zero and divisibility by 5 with digits from {2,3,4,5,6} requires last digit 5... but the working instead uses the alternating-sum condition for 11: a+c=b+d±11, combined with divisibility by 5 constraints on the digit set.
Working through valid combinations (as in the source): for each valid (a,b,c,d) satisfying a+c=b+5 pattern, 6 four-digit numbers are found to be divisible by 55, with no valid 5-digit number in range.
\[\text{Total} = 6\]