Questions: Question 3, 4.4.4 List the first twenty counting numbers in base seven (The digits 0,1,2, ..., 6 are used in base seven) Choose the correct answer below. A. 1,2,3,4,10,11,12,13,14,20,21,22,23,24,30,31,32,33,34,40 B. 1,2,3,4,5,6,10,11,12,13,14,15,16,20,21,22,23,24,25,26 C. 1,2,3,4,5,6,7,8,9, A, B, 10,11,12,13,14,15,16,17,18 D. 1,2,3,4,5,6,7,8,9, A, B, C, D, 10,11,12,13,14,15,16

Question 3, 4.4.4 List the first twenty counting numbers in base seven (The digits 0,1,2, ..., 6 are used in base seven) Choose the correct answer below. A. 1,2,3,4,10,11,12,13,14,20,21,22,23,24,30,31,32,33,34,40 B. 1,2,3,4,5,6,10,11,12,13,14,15,16,20,21,22,23,24,25,26 C. 1,2,3,4,5,6,7,8,9, A, B, 10,11,12,13,14,15,16,17,18 D. 1,2,3,4,5,6,7,8,9, A, B, C, D, 10,11,12,13,14,15,16
Transcript text: Question 3, 4.4.4 List the first twenty counting numbers in base seven (The digits $0,1,2, \ldots, 6$ are used in base seven) Choose the correct answer below. A. $1,2,3,4,10,11,12,13,14,20,21,22,23,24,30,31,32,33,34,40$ B. $1,2,3,4,5,6,10,11,12,13,14,15,16,20,21,22,23,24,25,26$ C. $1,2,3,4,5,6,7,8,9, A, B, 10,11,12,13,14,15,16,17,18$ D. $1,2,3,4,5,6,7,8,9, A, B, C, D, 10,11,12,13,14,15,16$
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Solution

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Solution Steps

Step 1: Understand the Base Seven System

In base seven, the digits used are \(0, 1, 2, 3, 4, 5, 6\). The counting numbers in base seven start from \(1\) and increment by \(1\) each time, just like in base ten, but once we reach \(6\), the next number is \(10\) (which represents \(7\) in base ten).

Step 2: List the First Twenty Counting Numbers in Base Seven

The first twenty counting numbers in base seven are: \[ 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 26 \]

Step 3: Compare with the Given Options

  • Option A: \(1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40\)
    This option skips \(5\) and \(6\) and incorrectly jumps to \(10\) after \(4\).

  • Option B: \(1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 26\)
    This option correctly lists the first twenty counting numbers in base seven.

  • Option C: \(1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10, 11, 12, 13, 14, 15, 16, 17, 18\)
    This option includes digits \(7, 8, 9, A, B\), which are not valid in base seven.

  • Option D: \(1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 10, 11, 12, 13, 14, 15, 16\)
    This option also includes invalid digits \(7, 8, 9, A, B, C, D\).

Final Answer

The correct answer is B.

\[ \boxed{\text{B}} \]

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