## Task 4 - 170724

Determine all four-digit natural numbers that are divisible by 24 and whose digit representation has the form $9 x 7 y$!

Here, $x$ and $y$ are to be replaced by one of the ten digits $(0, \ldots, 9)$ each.
If a four-digit number has the required properties, it follows first that it is divisible by 4, and thus (according to the divisibility rule for 4) the two-digit number with the digit representation $7 y$ is also divisible by 4.

Of the numbers $70, \ldots, 79$, only 72 and 76 are divisible by 4, so the only possibilities left are $y = 2$ and $y = 6$.

Now it follows further: If $y=2$, then for $x$, due to the divisibility rule for 3, only the digits 0, 3, 6, and 9 are possible. However, 3 and 9 are excluded due to the divisibility rules for 8, since 372 and 972 are not divisible by 8.

If $y=6$, then for $x$, due to the divisibility rules for 3, only the digits 2, 5, and 8 are possible. However, 2 and 8 are excluded, since 276 and 876 are not divisible by 8.

Therefore, only the numbers 9072, 9672, and 9576 can have the required properties.

These numbers do have these properties; because their digit representation is of the prescribed form, and they are divisible by 24.

Solutions of the 2nd Round 1977 taken from [5]

### 4.19.3 3rd Round 1977, Class 7