Question

Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting position?

$405$

$407$

$409$

$411$

$413$

Answer: D

Concepts:Diophantine Equation modular arithmetic optimization

Difficulty rating: 1740

Video solution:

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Written solution:

Let $r$ be the number of right jumps and $l$ be the number of left jumps. We need $5r-3l=2023.$ Modulo $5$ , this gives $-3l\equiv3\pmod5$ , so $l\equiv4\pmod5$ .

To minimize the total number of jumps, use the smallest possible $l$ , namely $l=4$ . Then $5r-12=2023$ , so $r=407$ .

The fewest number of jumps is $407+4=411$ .

Thus, D is the correct answer.

2023 AMC 8 Problem 18

Answer: D

Difficulty rating: 1740

Video solution:

Written solution:

Problem 18 in Other Years