A "dot" (commonly also called a "lattice point") is a point with integer coordinates.
In the plane, what is the total number of dots inside or on the boundary of the triangle with vertices (0,0), (x,0), (x,y) where x and y are positive integers?
In the event that it is not utterly obvious from the form of your answer that a whole number is being specified, give an independent argument to show this.
What total do you get if you count the three vertex dots together as just half a dot and any other boundary dots as half a dot each?
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Submitted by Richard
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Rating: 4.5000 (2 votes)
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Solution:
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The rectangle with vertices (0,0), (x,0), (x,y), and (0,y) contains (x+1)(y+1) dots, and it contains two copies of the given triangle which share the same hypotenuse. Thus
2N - H = (x+1)(y+1)
where N is the number of dots that we are trying to find, and H is the number of dots on the hypotenuse. Now a dot is on the hypotenuse if and only if its coordinates (m,n) satisfy n/m=y/x. Thus nx and my are common multiples of x and y and therefore nx and my are each the same multiple of the least common multiple of x and y, namely xy/d, where d is the greatest common divisor of x and y. Hence nx=my=kxy/d so that n=ky/d and m=kx/d. The values of k that work are then 0, 1, ..., d so that H=1+d. Hence the formula
N = [1+d+(x+1)(y+1)]/2.
We may independently verify that this is a whole number by noting that if x and y have the same parity then d has that same parity as well, but if x and y have different parity then d has odd parity.
When the boundary dots are treated partially as described, the total then comes out to be exactly the area of the triangle. |
