By a unit fraction, we mean a rational number with a numerator of 1. Symbolically, 1 b where b is a nonzero integer. Find six distinct, positive unit fractions whose sum is 1.
1/6 + 1/3 + 1/2 = 1, so
1/9 + 1/18 = 1/6
1/8 + 1/24 = 1/6
1/24 + 1/18 + 1/9 + 1/8 + 1/6 + 1/2 = 1
To find six distinct, positive unit fractions whose sum is 1, we can follow these steps:
1. Start by listing six distinct positive integers, which will be the denominators of the unit fractions. Let's use 2, 3, 4, 5, 6, and 7.
2. Set up an equation to represent the sum of the unit fractions:
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 1
3. To solve the equation, we need to find a common denominator for all the fractions. In this case, the least common multiple (LCM) of 2, 3, 4, 5, 6, and 7 is 420.
4. Now, we can rewrite each fraction with the common denominator of 420:
(210/420) + (140/420) + (105/420) + (84/420) + (70/420) + (60/420) = 1
5. Simplify the fractions by dividing both the numerator and the denominator by their greatest common divisor:
1/2 + 2/6 + 1/4 + 4/20 + 1/6 + 6/42 = 1
Now, the fractions become:
210/420 + 140/420 + 105/420 + 84/420 + 70/420 + 60/420 = 1
which can be simplified to:
1/2 + 1/3 + 1/4 + 1/5 + 1/7 + 1/42 = 1
6. Therefore, the six distinct, positive unit fractions whose sum is 1 are 1/2, 1/3, 1/4, 1/5, 1/7, and 1/42.