Math

posted by .

Find the smallest positive integer d such that d=105m+216n, where m & n are integers.

  • Math -

    Compute the GCD of 105 and 216 using thr Euclidean algorithm. A linear combination of 105 and 216 of the form

    a 105 + b 216

    can be reprented by the vector:

    (a, b, a 105 + b 216)

    We start with the two vectors:

    v1 = (105, 0, 105)

    v2 = (0,216, 216)

    The integer part of 216/105 is 2.

    v2 - 2 v1 = (-210, 216, 6)

    We now define the new v2 to be the old v1 while the above vector becomes the new v1:

    v1 = (-210, 216, 6)

    v2 = (105, 0, 105)

    We now repeat the previus step.

    Integer part of 105/6 is 17.

    v2 - 17 v1 = (3675, -3672, 3)

    If you then would run another step, you would end up with a last component of zero, this means that the GCD is 3 and you have:

    3 = 3675*105 - 3672*216

    You can then add that last vector with zero last component to make the integers m and n positive.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math, algebra

    2a+2ab+2b I need a lot of help in this one. it says find two consecutive positive integers such that the sum of their square is 85. how would i do this one i have no clue i know what are positive integers.but i don't know how to figure …
  2. Math

    Let a be an integer, then there are integers X, Y such that aX+(a+1)Y=1. Find the smallest positive value of Y.
  3. math

    there are three consecutive positive integers such that the sum of the squares of the smallest two is 221. write and equation to find the three consecutive positive integers let x= the smallest integer
  4. Math

    Find the smallest positive integer n such that the equation 455x+1547y=50,000+n has a solution (x,y) where both x and y are integers
  5. math

    1. The largest of five consecutive integers is twice the smallest. Find the smallest integer. 2. When the sum of three consecutive integers is divided by 9 the result is 7. Find the three integers. 3. If each of three consecutive integers …
  6. la sallle

    There are 3 consecutive even integers such that the quotient obtained by dividing twice the largest integer by the smallest integer is three less than three-fifths of the second integer. What are the integers?
  7. algebra

    find three consecutive odd integers such that the sum of the middle and largest integer is 21 more than the smallest integer
  8. Math

    Find three consecutive even integers such that the sum of the smallest integer and twice the median integer is 20 more than the largest integer.
  9. maths

    the non- decreasing sequence of odd integers {a1, a2, a3, . . .} = {1,3,3,3,5,5,5,5,5,...} each positive odd integer k appears k times. it is a fact that there are integers b, c, and d such that, for all positive integers n, aƱ = …
  10. Algebra

    Solve the following algebraically using one variable: Find three consecutive even integers such that the product of the smallest integer and the middle integer is eight less than eleven times the larger integer.

More Similar Questions