## Task 4 - 240614

Rita multiplies a number $z$ by 9 and gets the result 111111111.

(a) Which number $z$ is it?

(b) Determine a number $x$ that has the following property!

If you multiply $x$ by the number $z$ determined in (a), then the product is a number written entirely with the digit 8 (in the usual decimal notation).

(c) Are there any other numbers besides the number $x$ determined in (b) that also have this property?

If this is the case, determine another such number!
(a) Since $111111111: 9=12345679$, $z=12345679$ is the number that, when multiplied by 9, gives 111111111.

(b) From $12345679 \cdot 9=111111111$ it follows that $12345679 \cdot 72=888888888$ (1)

Therefore, for example, the number $x=72$ has the required property that the number $z \cdot x$ is written entirely with the digit 8.

(c) From (1) it follows that $12345679 \cdot 72 \cdot 1000000001=888888888 \cdot 1000000001,$ (2)

i.e., $12345679 \cdot 72000000072=888888888888888888$. (3)

Thus, for example, the number 72000000072 also has the required property.

Solutions of the I. Round 1984 taken from [5]

### 3.25.2 II. Round 1984, Class 6