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Suppose that $b$ is a positive integer greater than or equal to $2.$ When $197$ is converted to base $b$, the resulting representation has $4$ digits. What is the number of possible values for $b$

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MrRoyal

Complete Question

Suppose that b is a positive integer greater than or equal to 2. When 197 is converted to base b, the resulting representation has 4 digits. What is the number of possible values for b?

The possible values of b are 4 and 5

How to determine the possible numbers of b?

The conditions are given as:

  • b is at least 2
  • When 197 in base b has 4 digits.

So, we start by converting 197 to base 2 and above.

This is done as follows:

197 to base 2

2 | 197

    98 R 1

    49 R 0

    24 R 1

    12 R 0

    6 R 0

    3 R 0

    1 R 1

    0 R 1

So, we have:

197 = 11000101 in base 2

11000101 has more than 4 digits.

This means that b cannot be 2

Using the above method of conversion, we have:

  • 197 = 21022 in base 3
  • 197 = 3011 in base 4
  • 197 = 1242 in base 5
  • 197 = 525 in base 6
  • 197 = 401 in base 7

See that as the base increases, the number of digits decreases.

The numbers in base 4 and 5 have 4 digits.

Hence, the possible values of b are 4 and 5

Read more about base conversions at:

https://brainly.com/question/17946394

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