Math Problem Statement
Show that if n is an integer then the final digit of n^4 must be 0,1,5,6
Solution
To show that if is an integer, then the final digit of must be , , , or , let's examine the possible values for modulo 10, which will determine the last digit of .
Step-by-Step Solution
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Consider the possible last digits of : Since is an integer, the last digit of could be any of the digits from to . Let's go through each possible last digit and compute to find the last digit of .
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Evaluate for each case:
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If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is . -
If :
, so the last digit is .
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Summarize the results: From the above calculations, we can see that the possible last digits of are:
Conclusion
Thus, if is an integer, the last digit of must indeed be one of , , , or .
Would you like further details on any part of this proof?
Here are 5 related questions:
- How would you prove the result if the power were instead of ?
- What is the last digit of for various values of ?
- Can we generalize this approach to find the last digit of for any integer ?
- Why do some numbers repeat more frequently as last digits in powers of integers?
- What other properties of numbers can we determine by looking at specific powers?
Tip: When solving problems about last digits, consider using modular arithmetic with modulus 10 to simplify the calculations.
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Number Theory
Exponentiation
Formulas
n^4 mod 10
Modular arithmetic properties
Theorems
Modulo 10 behavior of powers
Suitable Grade Level
Grades 7-9