Problems

If \(x\) is a positive integer, the value of the function \(\mathrm{d}(x)\) is the sum of the digits of \(x\) in base 10. For example, \(\mathrm{d}(249) = 2 + 4 + 9 = 15\). An \(n\)-digit positive integer \(x\) is written in the form \(\displaystyle\sum_{r=0}^{n-1} a_r \times 10^r\), where \(0 \leqslant a_r \leqslant 9\) for all \(0 \leqslant r \leqslant n-1\) and \(a_{n-1} > 0\).

1. Prove that \(x - \mathrm{d}(x)\) is non-negative and divisible by \(9\).
2. Prove that \(x - 44\mathrm{d}(x)\) is a multiple of \(9\) if and only if \(x\) is a multiple of \(9\). Suppose that \(x = 44\mathrm{d}(x)\). Show that if \(x\) has \(n\) digits, then \(x \leqslant 396n\) and \(x \geqslant 10^{n-1}\), and hence that \(n \leqslant 4\). Find a value of \(x\) for which \(x = 44\mathrm{d}(x)\). Show that there are no further values of \(x\) satisfying this equation.
3. Find a value of \(x\) for which \(x = 107\mathrm{d}\left(\mathrm{d}(x)\right)\). Show that there are no further values of \(x\) satisfying this equation.