Let n be a positive integer whose decomposition into prime factors has no repeat
ID: 3036918 • Letter: L
Question
Let n be a positive integer whose decomposition into prime factors has no repeated prime. Let B = {x/x is a divisor of n}. For example, if n = 21 = 3 middot 7, then B = {1, 3, 7, 21). Let the following operations be defined on B: x + y = lcm (x, y) x middot y = gcd(x, y) x' = n/x Then + and middot are binary operations on B and ' is a unary operation on B. a. For n = 21, find (i) 3 middot 7 (ii) 7 middot 21 (iii) 1 + 3 (iv) 3 + 21 (v) 3' b. Prove that the commutative, associative, and distributive properties hold for both + and middot. c. Find the value of the "0" element and the "1" element, then prove properties 4 and 5 for both + and middot. d. Consider a value for n whose decomposition has repeated primes. In particular, let n = 12 = 2 middot 2 middot 3. Prove that, using the above definitions for + and middot, it's not possible to define a complement for 6 in the set {1, 2, 3, 4, 6, 12} Therefore a algebra cannot be constructed with n = 12 using the process described.Explanation / Answer
a . i . 3.7 = gcd (3,7) =1 as 3,7 are relatively prime nos
ii . 7 .21 = gcd(7,21 ) =7 ( 7is the highest common divisor of 7,21 0
iii. 1+3 = lcm (1,3) =3
iv . 3+21 = lcm (3,21 ) = 21
b . i . x+y = lcm (x,y ) = lcm (y ,x) = y+x => the operation + is commutative
ii. x . y = gcd 9x,y _ = gcd(y,x) = y .x => the operation . is commutative
iii. consider x+(y+z) = lcm ( x, y+z) = lcm ( x, (lcm (y,z) )= lcm (x,y,z)
= lcm(x+y,z). => + is associative
iv similarly we can prove . is also associtive
v . distiributive property : x.( Y+z) = gcd ( x, y+z) = gcd[ x, lcm (y,z) ]= lcm [ gcd(x,y) , gcd (x,z) ]
= x.y+x.z
c 'o ' element = 1 as lcm (1,x) =x
'1' elemtn ie the unit element wrt 2nd operation is 3 . x = gcd (3,x) =3 does nt exist
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