Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily  distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?
1. **Restate the problem in a more manageable form:**
   We need to prove that the number \(1\) can be represented as a sum of a finite number \(n\) of real numbers, each less than \(1\), and each containing only the digits \(0\) and \(7\) in their decimal representation. We also need to find the least possible value of \(n\).

2. **Reformulate the problem using a known rational number:**
   Consider the rational number \(\frac{1}{7} = 0.\overline{142857}\). We need to express \(\frac{1}{7}\) as a sum of \(n\) real numbers, each less than \(\frac{1}{7}\) and containing only the digits \(0\) and \(1\) in their decimal representation.

3. **Assume \(n < 10\) and find the least value of \(n\):**
   We assume that all the real numbers are positive and that \(n < 10\). The least value of \(n\) is found to be \(8\), for which we have the identity:
   \[
   \frac{1}{7} = 0.\overline{1} + 0.\overline{011111} + 0.\overline{010111} + 0.\overline{010111} + 0.\overline{000111} + 0.\overline{000101} + 0.\overline{000101} + 0.\overline{000100}.
   \]

4. **Consider the case with both positive and negative real numbers:**
   Suppose now that \(\frac{1}{7}\) is the sum of \(i > 0\) positive real numbers and \(j > 0\) negative real numbers, with \(i + j \leq 7\). Denote by \(u = 0.u_1u_2\ldots\) (respectively by \(-v = -0.v_1v_2\ldots\)) the sum of the positive (respectively negative) real numbers, so that \(\frac{1}{7} + v = u\). Note that \(u\) and \(v\) are necessarily strictly less than \(1\).

5. **Analyze the constraints on \(u\) and \(v\):**
   Since \(i, j < 7\), it follows that \(v_6 \geq 3\), so that \(j \geq 3\) and \(i \leq 4\). Then, we must have \(v_5 \geq 4\), so that \(j \geq 4\) and \(i \leq 3\). In this case, we find \(u_3 \geq 3\) and thus \(i = 3\). This finally implies that \(v_2 \geq 6\), which is impossible since \(j = 4\).

6. **Conclude the least value of \(n\):**
   The least value of \(n\) is therefore \(8\).

The final answer is \( \boxed{ 8 } \).