## 9-Digit Numbers

1. Let R(X) be the number created by reversing the digits of X.  When a 9-digit Palindromic number P is divided by a 7-digit number N, a remainder of R(N) is obtained.  Also, dividing N by R(N) gives a remainder D, where D is a prime number.  The last digit of neither P nor N is zero.  Find any triplets (P,N,D) satisfying these conditions.  If in addition, the sum of the digits of D is a perfect square, how many triplets exist?

2. Determine ten different nine digit decimal numbers beginning with the same digit, such that their sum is divisible by nine of the said numbers. How many possible solutions are there?

3. Determine four different nine digit decimal numbers beginning with the same digit, such that their sum is divisible by three of the said numbers. How many possible solutions are there?

4. Extension: Determine M+1 different N-digit decimal numbers beginning with the same digit, such that their sum is divisible by M of the said numbers, and N is as small as possible.  For which M<10 do solutions exist?

Source: Adapted from K Sengupta of India.  He does not yet have solutions for all of these.  Original Extension.

Solution
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