4. A frog jumps over lily pads. It can jump 7 lily pads to the right or 4 lily pads to the left. What is the minimum number of jumps needed for the frog to move exactly 2024 lily pads to the right?

![](https://cdn.mathpix.com/cropped/2024_06_03_1d05bbb73aad96dd359ag-3.jpg?height=285&width=1514&top_left_y=1945&top_left_x=294)
First solution.

Let $a$ be the number of jumps to the right and $b$ be the number of jumps to the left.

Then $7a - 4b = 2024$ for natural numbers $a$ and $b$, i.e., 1 POINT

$7a = 2024 + 4b = 4 \cdot (506 + b), \quad 3$ POINTS

so 7 must divide $506 + b$. 1 POINT

Since 506 gives a remainder of 2 when divided by 7, $b$ when divided by 7 must give a remainder of 5. 2 POINTS We want the smallest $b$ and the smallest $a$, so we take $b = 5$ and get $a = 292$. 2 POINTS The total number of jumps needed is at least $292 + 5 = 297$. 1 POINT

TOTAL 10 POINTS

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