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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 .

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Jan 18, 2012
Puzzle - Ordered Triples
by: Staff


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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


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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}

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