N is a 4-digit number and let M = S(N/10)* 10⁴ + floor(N/10), where:
S(x) = x - floor(x).

Find all possible values of N such that:
N - M = 3105.

M = 1000*N - 10000*floor(N/10) + floor (N/10)

So N - M = N - [1000*N - 10000*floor(N/10) + floor(N/10)] = 9999*floor(N/10) - 999N = 3105

Dividing both sides by 9 gives 1111*floor(N/10) - 111N = 345

If N = 0 mod 10, then floor(N/10) = N/10, so we have 1111*N/10 - 111N = .1N = 345, so N = 3450 is a solution.

If we increment N by 1, then the floor(N/10) part remains unchanged, but the -111N part contributes an extra -111 to the LHS of the equation, so the RHS decreases by 111.

Once we reach the next value of N that is 0 mod 10, we add another 1111 to the LHS.  The net result is 10 * (-111) + 1111 = 1.  So incrementing N by 10 increments the RHS of the equation by 1.

So to make the RHS = 345, we can add 1111 to N (the first 1110 will add 111 to the RHS, and then the last 1 will deduct 111 from the RHS).

So additional solutions are 4561, 5672, 6783, 7894, and 9005.

 

Posted by tomarken on 2014-07-07 15:38:04