3-5. How many positive integers $x$, less than 10000, are there for which $2^{x}-x^{2}$ is divisible by 7?
Solve 5. Answer: 2857. The remainders of the division by 7 of the numbers $2^{x}$ and $x^{2}$ repeat with periods of 3 and 7, respectively, so the remainders of the division by 7 of the numbers $2^{x} - x^{2}$ repeat with a period of 21. Among the numbers $x$ from 1 to 21, the numbers that give equal remainders from the division by 7 of the numbers $2^{x}$ and $x^{2}$ are exactly 6 numbers. Therefore, among the numbers from 1 to $9996 = 7 \cdot 476$, there are $476 \cdot 6 = 2856$ required numbers. Direct verification using the obtained sequence of remainders shows that among the remaining numbers 9997, 9998, and 9999, only the number 9998 has the required property.

## VII Olympiad (1941)

## First Round

## 7 - 8 grades