Divisbility

Prove that for all odd integers n, (1^n)+(2^n)+(3^n)…+(n^n) is a proper multiple of 1+2+3+…n

number theory

write the numbers 25, 32, 56 to the base 5. write 47, 68, 127 to the base 2.

Euclid's Division Lemma and Fundamental Theorem of Arithmetic

1. Without assuming Theorem 2-1, prove that for each pair of integers j and k (k > 0), there exists some integer q for which j — qk is positive. 2. The principle of mathematical induction is equivalent to the following statement, called the least-integer principle: Every non-empty set of positive integers has a least element. ...continues

divisibility

find the greatest common divisor of the following pairs: a)527, 765 (use technique like 527=341*1+186) b)361, 1178 (use technique like 527=341*1+186) c) -find the gcd (d) of 299, 481 (use technique like 31=186-155*1 --> 31=186-(341-186*1) -find integers such that 299x+481y=d -now replace 299 and 481 by 129 and 301

Fundamental Theorem of Arithemtic : Lowest Common Multiples and Diophantine Equations

Please solve the following problems: 1. Compute the following ... 2. Let Fm be the set of all integral multiples of the integer m. Prove that ... 3. Draw the graphs of the straight lines defined by the following Diophantine equations ... 4. Prove that every integer is uniquely representable as the product of a non-negati ...continues

Residue Systems : Modulo

5. If m = 11, then a reduced residue system modulo m is 1,2,3,4,5,6,7,8,9,10. Exhibit the pairing of each of the preceding numbers with its inverse modulo m (like Chinese remainder theorem). 7. What is the remainder when 41^5 is divided by 3? When 473^38 is divided by 5? 8. Prove that if p is a prime congruent to 1 modulo ...continues

Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality ... holds true for all x E R then .... Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and co ...continues

Congruences : Prime Congruences, Systems of Congruences and Polynomials

1. If p is a prime congruent to 1 modulo 4 then ((p-1)/2)!^2 ≡ -1 mod p. Use this to find solutions to the following congruences: x^2 ≡-1(mod13) x≡-1(mod7) 2. Prove that for each odd prime p and .... 3. Find all solutions of each of the following systems of congruences: 4. F ...continues

congruences

5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.

1. If gcd(m,n) = 1, then φ(m,n) = φ(m)φ(n). Use this to give a proof that φ(n) = n Π(1 – 1/p) p/n 2. Prove that d(n) is odd iff n is a perfect square. 3. Prove that σ(n) ≡ d(m)(mod 2) where m is the largest odd factor of n. 3.(2nd Part) If σ(n) = 2n, n is a perfect number. Prove that if n is a perfect number , then ∑1/d = 2. d/n 4.Evaluate σ(210), φ(100) and σ(999). 5.Evaluate d(47), d(63) and d(150).

Arithmetic Functions Combinatorial Study of φ(n) 1. If gcd(m,n) = 1, then φ(m,n) = φ(m)φ(n). Use this to give a proof that φ(n) = n Π(1 – 1/p ...continues