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 nn is an integer, then the final digit of n4n^4 must be 00, 11, 55, or 66, let's examine the possible values for n4n^4 modulo 10, which will determine the last digit of n4n^4.

Step-by-Step Solution

  1. Consider the possible last digits of nn: Since nn is an integer, the last digit of nn could be any of the digits from 00 to 99. Let's go through each possible last digit and compute n4mod10n^4 \mod 10 to find the last digit of n4n^4.

  2. Evaluate n4mod10n^4 \mod 10 for each case:

    • If n0(mod10)n \equiv 0 \pmod{10}:
      n404=0(mod10)n^4 \equiv 0^4 = 0 \pmod{10}, so the last digit is 00.

    • If n1(mod10)n \equiv 1 \pmod{10}:
      n414=1(mod10)n^4 \equiv 1^4 = 1 \pmod{10}, so the last digit is 11.

    • If n2(mod10)n \equiv 2 \pmod{10}:
      n424=166(mod10)n^4 \equiv 2^4 = 16 \equiv 6 \pmod{10}, so the last digit is 66.

    • If n3(mod10)n \equiv 3 \pmod{10}:
      n434=811(mod10)n^4 \equiv 3^4 = 81 \equiv 1 \pmod{10}, so the last digit is 11.

    • If n4(mod10)n \equiv 4 \pmod{10}:
      n444=2566(mod10)n^4 \equiv 4^4 = 256 \equiv 6 \pmod{10}, so the last digit is 66.

    • If n5(mod10)n \equiv 5 \pmod{10}:
      n454=6255(mod10)n^4 \equiv 5^4 = 625 \equiv 5 \pmod{10}, so the last digit is 55.

    • If n6(mod10)n \equiv 6 \pmod{10}:
      n464=12966(mod10)n^4 \equiv 6^4 = 1296 \equiv 6 \pmod{10}, so the last digit is 66.

    • If n7(mod10)n \equiv 7 \pmod{10}:
      n474=24011(mod10)n^4 \equiv 7^4 = 2401 \equiv 1 \pmod{10}, so the last digit is 11.

    • If n8(mod10)n \equiv 8 \pmod{10}:
      n484=40966(mod10)n^4 \equiv 8^4 = 4096 \equiv 6 \pmod{10}, so the last digit is 66.

    • If n9(mod10)n \equiv 9 \pmod{10}:
      n494=65611(mod10)n^4 \equiv 9^4 = 6561 \equiv 1 \pmod{10}, so the last digit is 11.

  3. Summarize the results: From the above calculations, we can see that the possible last digits of n4n^4 are: 0,1,5, or 6.0, 1, 5, \text{ or } 6.

Conclusion

Thus, if nn is an integer, the last digit of n4n^4 must indeed be one of 00, 11, 55, or 66.

Would you like further details on any part of this proof?

Here are 5 related questions:

  1. How would you prove the result if the power were n3n^3 instead of n4n^4?
  2. What is the last digit of n5n^5 for various values of nn?
  3. Can we generalize this approach to find the last digit of nkn^k for any integer kk?
  4. Why do some numbers repeat more frequently as last digits in powers of integers?
  5. 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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