Math 290 Exam questions and answers with detailed accurate solutions with a grade of A+

A property of binomial coefficients - ANSC(n, k) + C(n, k+1) = C(n+1, k+1) Condition for C(n, k) + C(n, k+1) = C(n+1, k+1) - ANSn = 0 Binomial Theorem - ANS(x + y)^n = sum( x^(n - k) y^k C(n, k) ) from k=0 to n Binomial Theorem conditions - ANSn = 0 The division algorithm - ANSn =qd+r with 0 ≤ r |d| and q and r are unique integers conditions for the division algorithm - ANSn, d integers and d is non-zero GCD-switching Theorem - ANSLet a,b,c,x ∈ Z and assume that a = xb+c. Then GCD(a,b) = GCD(b,c). GCD-switching Theorem conditions - ANSintegers that are set up like the division algorithm, BUT NOT SAME conditions as division algorithm What does the euclidean algorithm do - ANScompute the GCD Euclidean Algorithm Setup - ANSGiven two integers a,b not both 0, assume that a= 0 and that |a| ≥ |b| (if either of these does not hold, swap a and b so that both hold). Euclidean algorithm for GCD(a,b) where b=0 - ANSthen GCD(a, 0) = |a| Euclidean algorithm where first r is zero - ANSthen we know b divides a, so |b| is the GCD size of numbers where one divides the other - ANSif a, b are NONZERO and a|b then |a| |b| if two numbers divide each other - ANSmust be NONZERO. if a|b and b|a then a = +/-b (or you could say that |a| = |b| ) how many integers divide a number - ANSNONZERO integer, then the quantity is finite Common Divisior - ANSc|a and c|b and all are integers.