LCM and GCD proof.

Related BrainMass Solutions

• Discrete Math : History, Appearance and Application of Chinese Remainder Theorem in Chinese and Hindu Writings

  Both have exactly the same gcd and lcm but obviously different products. On the other hand, both gcd and lcm are associative: gcd(m1, (gcd(m2, m3)) = gcd(gcd(m1, m2), m3) and, both equal gcd(m1, m2, m3).

• Least common multiples

  integers, then ab=gcd(a,b)lcm(a,b).

• Systems of Congruences : Simultaneous Solutions

  Prove that gcd (a, lcm[b, c]) = lcm[gcd(a,b), gcd(a,c)]. 2.

• Euclidean Algorithm

  Also find the gcd by Euclid's algorithm and then find the lcm from lcm(x,y)=xy/gcd(x,y). 3. Prove the following statment by induction. Use the headings Base, Inductive step, and By induction.

• Rational Numbers

  ; // denominator lcm = (d1 * d2 ) / GCD (d1, d2) ; int num ; num = (lcm / d1) * n1 + (lcm / d2) * n2 ; if (num < 0) { num = -num ; sign = -1 ; } else sign = 1 ; rational rat ; rat = new_rat (num, lcm) ;

• Cyclic Groups

  a) Proof: We only need to show that a^i has order m. Suppose a^i has order n, then we have (a^i)^n=a^(in)=a^m=1, then m|(in), but gcd(m,i)=1 since i and m are relatively prime, then we must have m|n.

• Permutations and Combinations

  Answer: c is the greated common divisor of a and b. i.e., c = gcd(a, b). d is the least common multiple of a and b. i.e., d = lcm(a, b). b. Proof: Let T = R intersect S.

• Computing greatest common divisor and least common multiple

  It is not possible to do so because their GCD is 3, so their Least Common Multiple (LCM) is 9*15/3=45 which is not 225. The solution gives detailed steps on computing a greatest common divisor and a least common multiple.

• Euler Function

  (In other words, if f(k)=(a,b), then gcd(k,mn)=1 if and only if gcd(a,m)=1 and gcd(b,n)=1) c. Use the results of the preceeding two exercises to complete the proof of the conjecture.