If P713 is divisible by 11, find the value of the smallest natural number P?

Concept: A number is divisible by 11 when the difference between the sum of its digits at the odd positions and the sum of its digits at the even positions (counted from the left) is either 0 or a multiple of 11.

Applying this to P713:

1. The digits of P713 in order are: 1st position = P, 2nd position = 7, 3rd position = 1, 4th position = 3.

2. Sum of digits at the odd positions (1st, 3rd) = P + 1. Sum of digits at the even positions (2nd, 4th) = 7 + 3 = 10.

3. For divisibility by 11, the difference (P + 1) - 10 must equal 0 or a multiple of 11.

4. Solving (P + 1) - 10 = 0 gives P = 9. Since P must be a single natural-number digit (0-9), P = 9 is the only value that works.

Cross-check: substituting P = 9 gives the number 9713, and 9713 ÷ 11 = 883 exactly, confirming the number is divisible by 11.

Hence, the value of P is 9.