Answer to Question #74778 in Combinatorics | Number Theory for Daryl Ligutan

Question

1. Use mathematical induction to show that 8 │ (5^2n + 7).
Hint: 5^2(k+1) + 7 = 5^2(5^2k + 7) +(7 - 5^2·7)

2. Use the Division Algorithm to establish that 3a^2 – 1 is never a perfect square.

3. Use the Euclidean Algorithm to obtain integers x and y satisfying
gcd(1769,2378) = 1769x + 2378y.

4. Determine all solutions in the positive integers of the following Diophantine equation:
123x + 360y = 99.

5. Find the prime factorization of integer 1234, 10140, and 36000.

6. Give an example of a^2 ≡ b^2 (mod n) need not imply a ≡ b (mod n).

7. Show the following statements are true:
a. For any integer a, the unit digit of a^2 is 0, 1, 4, 5, 6 or 9.
b. Any one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occur as the units digit of a^3.
c. For any inter a, the units digit of a^4 is 0, 1, 5 or 6.

Answer 1

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