# 6 Digit Number Puzzle

## 6 Digit Number Puzzle

How many 6-digit numbers can be formed by using the digits 1 to 6 without repetition. Such that the number is divisible by the digit at its unit place? ## Divisibility Rule Analysis

Divisibility Rule of 1 Divisibility Rule of 2 Divisibility Rule of 3 Divisibility Rule of 4 Divisibility Rule of 5 Divisibility Rule of 6 #### Detailed Analysis

Explanation-
XXXXX1 is always divisible by 1, so we have 5! numbers.
XXXXX2 is always divisible by 2, so we have 5! numbers.
XXXXX3 is always divisible by 3 (sum of digits is always 21), so we have 5! numbers.
XXXXY4 is divisible by 4 only if Y is 2 or 6, so we have $2 \times 4!$ numbers.
XXXXX5 is always divisible by 5, so we have 5! numbers.
XXXXX6 is always divisible by 6 (even number divisible by 3), so we have 5! numbers.

So total number of numbers with required property

= $5 \times 5! + 2\times 4! = 600 + 48 = 648$ numbers.  #### Most Asked Puzzles

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