Yes, 1 is a powerful number according to one specific definition.
Understanding Powerful Numbers
The term "powerful number" can refer to different concepts in number theory. However, one specific definition, as mentioned in the provided reference, defines a powerful number as a number which is the sum of any positive powers of its digits. Here's a breakdown:
Powerful Numbers Defined
- A number is considered "powerful" if it can be expressed as the sum of its digits raised to some positive integer power (not necessarily the same power for each digit).
- This definition focuses on the relationship between a number and the powers of its individual digits.
Determining if 1 is a powerful number:
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The number 1 consists of a single digit, '1'.
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If we raise the digit 1 to any positive integer power (e.g., 11, 12, 13), the result will always be 1.
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Therefore, 1 can be expressed as the sum of its digit (which is just 1 itself) raised to a positive power.
Example: 1 = 11
Examples of Other Powerful Numbers:
As provided in the reference, some examples of numbers following this specific definition of powerful numbers include:
- 2 ( 2 = 21)
- 3 ( 3 = 31)
- 4 ( 4 = 41)
- 5 ( 5 = 51)
- 6 ( 6 = 61)
- 7 ( 7 = 71)
- 8 ( 8 = 81)
- 9 ( 9 = 91)
- 24 (24 = 24 + 41)
- 43 (43 = 42 + 33)
- 63 (63 = 62 + 33)
- 89 (89 = 82 + 91)
Key Takeaway:
Based on this definition, 1 fits the criteria because it is equivalent to its own digit raised to the power of 1.
