Question

Which of the following numbers has the greatest number of unique digits?

• A. 1/6
• B. 1/4
• C. 1/3
• D. 3/4
• E. 5/7

Hint:

For a fraction \(p/q\) in simplest form, the length of the repeating part of its decimal expansion is related to the prime factors of the denominator \(q\). If \(q\) is a prime number (other than 2 or 5), the length of the repeating cycle can be up to \(q-1\). For 5/7, the length is 6, which is \(7-1\). This often leads to a large number of unique digits.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The question asks to find which fraction, when converted to a decimal, uses the highest number of different (unique) digits. This requires converting each fraction to its decimal representation and then counting the unique digits that appear.

Step 2: Key Formula or Approach:
We will perform the division for each fraction to find its decimal form. Then we will list the set of unique digits for each and count them.

Step 3: Detailed Explanation:
Let's convert each fraction to a decimal and count its unique digits.
(A) 1/6
\[ 1 \div 6 = 0.1666... = 0.1\overline{6} \] The digits used are 1 and 6.
Unique digits: \{1, 6\}. Count = 2.
(B) 1/4
\[ 1 \div 4 = 0.25 \] This is a terminating decimal. The digits used are 2 and 5.
Unique digits: \{2, 5\}. Count = 2.
(C) 1/3
\[ 1 \div 3 = 0.333... = 0.\overline{3} \] The only digit used is 3.
Unique digits: \{3\}. Count = 1.
(D) 3/4
\[ 3 \div 4 = 0.75 \] This is a terminating decimal. The digits used are 7 and 5.
Unique digits: \{5, 7\}. Count = 2.
(E) 5/7
To convert 5/7, we perform long division: \[ 5 \div 7 = 0.714285714285... = 0.\overline{714285} \] The repeating block of digits is 714285.
The digits used are 7, 1, 4, 2, 8, and 5.
Unique digits: \{1, 2, 4, 5, 7, 8\}. Count = 6.
Comparing the counts of unique digits: \[\begin{array}{rl} \bullet & \text{1/6: 2} \\ \bullet & \text{1/4: 2} \\ \bullet & \text{1/3: 1} \\ \bullet & \text{3/4: 2} \\ \bullet & \text{5/7: 6} \\ \end{array}\] The fraction 5/7 has the greatest number of unique digits.
Step 4: Final Answer
The number with the greatest number of unique digits is 5/7.