Exercise 1644Number theory › Divisibility and gcdOlympiadAdded 3 Nov 2026

Egyptian fractions

#Egyptian fractions#unit fractions#greedy algorithm

Statement

Express $\frac{5}{7}$ as a sum of distinct unit fractions $\frac1{n_1}+\cdots+\frac1{n_k}$ with $n_1<\cdots<n_k$ integers.

Hint

Use the greedy algorithm: subtract the largest unit fraction $\le 5/7$ .

Show solutionHide solution

Step 1. $\lceil 7/5\rceil=2$ , so $1/2\le5/7$ . Subtract: $5/7-1/2=10/14-7/14=3/14$ .

Step 2. $\lceil14/3\rceil=5$ , so $1/5\le3/14$ . Subtract: $3/14-1/5=15/70-14/70=1/70$ .

Step 3. $1/70$ is already a unit fraction.

Conclusion. $5/7=1/2+1/5+1/70$ .

Check. $1/2+1/5+1/70=(35+14+1)/70=50/70=5/7$ . Correct.