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Combinatorics | Number Theory
Find all positive integers n such that n^4 - 1 is divisible by 5
Combinatorics | Number Theory
Find all positive integers n such that n^4-1 is divisible by 5.
Combinatorics | Number Theory
Find all positive integers n such that n^4 -1
is divisible by 5.
is divisible by 5.
Combinatorics | Number Theory
Calculate GCD(8,35) using EA (Euclid's Algorithm) and calculate the multiplicative inverse of 8 E Z3,, if it exists Also try the same for 21 E Z35.
Combinatorics | Number Theory
Prove that there are infinitely many prime numbers of the form 6n + 5. Hint: any prime number p > 3 has the form p = 6n + 1 or p = 6n + 5 for some integer n. Use this fact and the fact that the product of two numbers of the form 6n + 1 has the same form.
Combinatorics | Number Theory
Tom, Michael and Jane visit the same sport club. Tom visits the club every 5th day, Michael every 6th day and Jane every 8th day. If all three of them were in the sport club on August 5th, Sunday, what will be the next day when all of them will be there?
Combinatorics | Number Theory
Let a = 3333333333^7777777777 (yes, there are ten digits ”3” and ten digits ”7”). If a is
written in decimal notation, what are the last two digits?
Hint: You need to compute a mod 100. Use Euler’s theorem and the fact that φ(100) = 40.
written in decimal notation, what are the last two digits?
Hint: You need to compute a mod 100. Use Euler’s theorem and the fact that φ(100) = 40.
Combinatorics | Number Theory
Let p and q be distinct prime numbers. Prove that p
q−1 + q
p−1 ≡ 1 mod pq.
q−1 + q
p−1 ≡ 1 mod pq.
Combinatorics | Number Theory
Let p be an odd prime number, so that p = 2k + 1 for some positive integer k. Prove that
(k!)2 ≡ (−1)k+1 mod p.
Hint: Try to see how to group the terms in the product
(p − 1)! = (2k)! = 1 ∗ 2 ∗ 3 · · ·(2k − 2) ∗ (2k − 1) ∗ (2k)
to get two products, each equal to k! modulo p.
(k!)2 ≡ (−1)k+1 mod p.
Hint: Try to see how to group the terms in the product
(p − 1)! = (2k)! = 1 ∗ 2 ∗ 3 · · ·(2k − 2) ∗ (2k − 1) ∗ (2k)
to get two products, each equal to k! modulo p.
Combinatorics | Number Theory
Part 1: Prove that the system of linear congruences in one variable
x ≡ b1 mod m1
x ≡ b2 mod m2
is solvable if and only if (m1, m2)|b1 − b2. In this case prove that the solution is unique modulo [m1, m2].
Hint: follow the proof of the Chinese Remainder Theorem.
Part 2: Solve the system of congruences
x ≡ 3 mod 4
x ≡ 1 mod 6
x ≡ b1 mod m1
x ≡ b2 mod m2
is solvable if and only if (m1, m2)|b1 − b2. In this case prove that the solution is unique modulo [m1, m2].
Hint: follow the proof of the Chinese Remainder Theorem.
Part 2: Solve the system of congruences
x ≡ 3 mod 4
x ≡ 1 mod 6
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