# a is the factor of ordered triples

by Andrew
(New York)

Find the number of ordered triples (a,b,c), where a, b, and c are positive integers, a is a factor of b, a is a factor of c, and a + b + c = 100 .

### Comments for a is the factor of ordered triples

 Jan 18, 2012 Puzzle - Ordered Triples by: Staff ---------------------------------------------------------- Part II Use trial and error calculations. Select a value of a₀ (either 1, 2, 3, or 5), select a value for b₁, and then calculate c₁. a₀*1 + a₀*b₁ + a₀*c₁ = 100 a₀ = a₀*1 , b₀ = a₀*b₁ , c₀ = a₀*c₁ 1st solution If a₀ = 1, and b₀ = 1, then c₀ = 98 a₀ + b₀ + c₀ = 100 1 + 1 + 98 = 100 1 is a factor of 1, and 1 is also a factor of 98 2nd solution If a₀ = 2, and b₀ = 4, then c₀ = 94 a₀ + b₀ + c₀ = 100 2 + 4 + 94 = 100 2 is a factor of 4, and 2 is also a factor of 94 3rd solution If a₀ = 4, and b₀ = 16, then c₀ = 80 a₀ + b₀ + c₀ = 100 4 + 16 + 80 = 100 4 is a factor of 16, and 4 is also a factor of 80 4th solution If a₀ = 5, and b₀ = 25, then c₀ = 70 a₀ + b₀ + c₀ = 100 5 + 25 + 70 = 100 5 is a factor of 25, and 5 is also a factor of 70 5th solution If a₀ = 5, and b₀ = 50, then c₀ = 45 a₀ + b₀ + c₀ = 100 5 + 50 + 45 = 100 5 is a factor of 50, and 5 is also a factor of 45 There are many other solutions which can be worked out by trial and error for the values of: a₀ ∈ {1,2,4,5} Thanks for writing. Staff www.solving-math-problems.com

 Jan 18, 2012 Puzzle - Ordered Triples by: Staff Part I Question: by Andrew (New York) Find the number of ordered triples (a,b,c), where a, b, and c are positive integers, a is a factor of b, a is a factor of c, and a + b + c = 100 . Answer: Ordered Triple: set of THREE NUMBERS (x,y,z) IN ORDER. An ordered triple will look like this: 5,-9,18. The order of the numbers does matter. The first number (5, , ) represents the “x” value, the middle number ( ,-9, ) represents the “y” value, and the third number ( , ,18) represents the “z” value. The ordered triple could stand for a point in a 3-dimensional coordinate system (just as an ordered pair x,y is used to represent a point on a 2-dimensional x-y coordinate plane), or be used for some other purpose. An ordered triple (x,y,z) is the solution to a system of three simultaneous equations: A₀x + B₀y + C₀z = D₀ A₁x + B₁y + C₁z = D₁ A₂x + B₂y + C₂z = D₂ ----------------------------------------------- For this particular problem, you do not have 3 equations. a₀ + b₀ + c₀ = 100 therefore: a₀*1 + a₀*b₁ + a₀*c₁ = 100 a₀(1 + b₁ + c₁) = 100 (1 + b₁ + c₁) = 100/a₀ 1 + b₁ + c₁ = 100/a₀ 1 - 1 + b₁ + c₁ = (100/a₀) - 1 0 + b₁ + c₁ = (100/a₀) - 1 b₁ + c₁ = (100/a₀) - 1 1. for every value of a₀ (an integer), b₁ + c₁ MUST BE ANOTHER INTEGER. 2. for every value of a₀ (an integer), (b₁ + c₁)* a₀ + a₀ = 100 Using trial and error calculations, there are only 8 possibilities which meet both of these conditions: a₀ = 1 b₁ + c₁ = 99 a₀ = 2 b₁ + c₁ = 49 a₀ = 4 b₁ + c₁ = 24 a₀ = 5 b₁ + c₁ = 19 a₀ = 10 b₁ + c₁ = 9 a₀ = 20 b₁ + c₁ = 4 a₀ = 25 b₁ + c₁ = 3 a₀ = 50 b₁ + c₁ = 1 3. a₀ must also be a factor of b₀ and c₀ There are only 4 possibilities which meet all three conditions: a₀ = 1 b₁ + c₁ = 99 a₀ = 2 b₁ + c₁ = 49 a₀ = 4 b₁ + c₁ = 24 a₀ = 5 b₁ + c₁ = 19 a₀ can only be 1, 2, 4, or 5 a₀ ∈ {1,2,4,5} ----------------------------------------------------------