divisibility of sum of integers raised to odd powers
Prove that for all odd integers n, (1^n)+(2^n)+(3^n)…+(n^n) is a proper multiple of 1+2+3+…n
write the numbers 25, 32, 56 to the base 5. write 47, 68, 127 to the base 2.
euclid's division lemma / fundamental theorem of arithmetic
please see attached problems 2 credits/problem theorem 2-1 is euclid's division lemma j=qk+r
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
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
Please answer the following questions: 1. Prove the binomial theorem ... 2. Prove that if p is a prime and n is a positive integer, then p / nP - n 3. Prove that x / (1-x)^2 is the generating function for the sequence a1=1, a2=2,..., an=1/n,... 4. Prove that -log (1-x) is the generating function for the sequence a1=1, a2=1 ...continues
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
Let n be a natural number... Please see attached.
If p is a prime congruent.... Please see attached.
5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.
