How many solutions does u1+u2+u3+u4 = 30 have in non-negative integers with u1 >=3, u2 <= 8, 3 <= u3 <=8, u4 >=0? Ok, my attempt. Let v1 = u1 - 3, v2 = u2, v3 = u3 - 3 and v4 = u4. Then we have v1+v2+v3+v4 = 24 with v1 >= 0, v2 <=8, v3 <= 5, v4 >=0. So without constraints there are C(24+4-1, 24) = C(21,24) solutions (is th ...continues
Binomial Coefficients Formula Proof
n Prove that if n>=2, then Σ (-1)^(r-1) r n!/(r!n-r!) = 0 r=1 I think it should be done by mathematical induction. From the examples, I’m assuming that mathematical induction should be used. If you are able, please use this method.
Fundamental Theorem of Arithmetic
Please show complete solution so that I may understand the reasoning behind each step. I will need a thorough understanding to complete future assignments. See attached file for full problem description.
1.66 Find all solutions x to each of the following congruences: (I) 3x 2 mod 5 (II) 7x 4 mod 10 (III) 243x + 17 101 mod 725 (IV) 4x + 3 4 mod 5 (V) 6x + 3 4 mod 10 (VI) 6x + 3 1 mod 10
Number Theory : Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.
Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.
Consider the congruence ax ≡ b mod m When (a,m)=d. Show that ax ≡ b mod m has a solution if and only if d | b.
(See attached file for full problem description with all symbols) --- 2.14 (I) Prove that an infinite set X is countable if and only if there is a sequence of all the elements of X which has no repetitions. (II) Prove that every subset S of a countable set X is itself countable. (III) Prove that if ...continues
(See attached file for full problem description) 2.22 Define f: {0,1,2,…,10} {0,1,2,…,10} by f(n)= the remainder after dividing by 11. (I) Show that f is a permutation. (II) Compute the parity of f. (III) Compute the inverse of f.
(See attached file for full problem description with all symbols) --- 2.34 (I) How many elements of order 2 are there in and in ? Show work. (Answer: 25, 75 respectively) (II) How many elements of order 2 are there in ?
Opposite, Odd and Even Numbers
1. The Greeks believed matter and energy were opposites. So is Einstein right in saying opposites are equivalent? 2. Can you add opposites? 3. You can add odd numbers and even numbers. Could they represent opposite theories like Plank and Maxwell? Can even and odd numbers be used as algebra in equations.
