# Thread: How many lattice points are in here?

1. ## How many lattice points are in here?

For x between -100 and 100, inclusive, in the equation

y= - 4/3 x + 70/3

2. ## Re: How many lattice points are in here?

Originally Posted by orange
For x between -100 and 100, inclusive, in the equation

y= - 4/3 x + 70/3
do you consider a lattice point to be a point {x,y} where x,y are integers? Lattices can be however you define them.

If so

how often will an x value produce an integer y value given the denominator of both terms in y(x)?

How many integer values of x are there for {-100 <= x <= 100}

So roughly how many lattice points will there be?

Now look at the edges of your x domain and make sure you don't have to subtract 1.

-4(-98)+70 = 462 which is divisible by 3, so they start at -98

-4(100)+70 = -330 which is divisible by 3 so they stop at 100

You can figure out how many there are from all this.

3. ## Re: How many lattice points are in here?

Yes! Where x and y are integers. Thanks!

4. ## Re: How many lattice points are in here?

I editted my reply a bit to fix a sign error. You might want to recheck it.

5. ## Re: How many lattice points are in here?

I'm thinking it's 32 + 33, but I get so confused on this kind of thing because, from 0-20, there are 7 x's that work, but if you just divide 20 by 3 you only get 6 with some remainder. If you divide 100 by 3 you have a remainder of 1, I don't think that's enough to get another number though?

6. ## Re: How many lattice points are in here?

Originally Posted by orange
I'm thinking it's 32 + 33, but I get so confused on this kind of thing because, from 0-20, there are 7 x's that work, but if you just divide 20 by 3 you only get 6 with some remainder. If you divide 100 by 3 you have a remainder of 1, I don't think that's enough to get another number though?
what is 100 - (-100) ?

7. ## Re: How many lattice points are in here?

Originally Posted by orange
For x between -100 and 100, inclusive, in the equation
y= - 4/3 x + 70/3
Note that we need $-4x+70$ to be a multiple of three.

That is true if $x=100$ and false if $x=-100$.

So it is true if $x=100-3k,~k=0,1,\cdots,~?$

8. ## Re: How many lattice points are in here?

Oh, I think I got a way - line up all the numbers (positives first) that work for x... you get 1, 4, 7, 10.... 100. Add 2 to everything and you get multiples of 3, then divide everything by three to get 1, 2, 3... etc. and you have the same AMOUNT of numbers. You end up with 34 for numbers up to 100 (102/3=34).

Same thing starting from -98 and ending at -3.... add -1 to each number to get -99, -96,..., -3 and divide by -3 to find there are 99/3= 33 numbers. I won't count 0 because 0+70 isn't divisible by 3.

So all in all we have 34 + 33?

9. ## Re: How many lattice points are in here?

that's the number I got but it's really not that tough. As I noted the first was at -98 and the last at 100.

so -98 + 3k = 100

k=66

but the point k=0 is included so we have 67 points total.

10. ## Re: How many lattice points are in here?

Darn... I forgot another condition, y is also between -100 and 100 inclusive.

11. ## Re: How many lattice points are in here?

So I guess x can be at most 92, at least -57, if I figured correctly..

12. ## Re: How many lattice points are in here?

Leaving -56 as the smallest one and 91 as the greatest one. So that subtracts 14 numbers from the bottom and 3 from the top?

13. ## Re: How many lattice points are in here?

Originally Posted by orange
Leaving -56 as the smallest one and 91 as the greatest one. So that subtracts 14 numbers from the bottom and 3 from the top?
you're killin me. Who cares what the y values are. You just need to find the number of x's so that -4x+70 is divisible by 3.

Once you find your starting point that's every 3rd x. We know the starting point. It's -98 given your range of -100,100.

so -98, -95, -93 ... 94, 97, 100

are your x points. Now just count them. Like I said before 100 = -98 + 3*66, but -98 is included as well so you have 67 total points.

14. ## Re: How many lattice points are in here?

Yeah, I meant in the original problem I was trying to solve, the stipulation was that y was also between -100 and 100! I forgot to include that rather vital tidbit when I asked it on here.

15. ## Re: How many lattice points are in here?

Hello, orange!

$\text{On }[\text{-}100, 100],\text{ how many lattice points are on the line: }y \:=\:-\tfrac{4}{3}x + \tfrac{70}{3}$

We have: . $y \:=\:\text{-}x - \frac{x}{3} + 23 + \frac{1}{3} \quad\Rightarrow\quad y \:=\:\text{-}x + 23 + \frac{1-x}{3}$

Since $y$ is an integer, then $1-x$ must be a multiple of 3.
. . That is: . $1 - x \:=\:3a\:\text{ for some integer }a$
We have: . $x \:=\:1-3a$

Since $x \ge \text{-}100\!:\;\;1-3a \,\ge\,\text{-}100 \quad\Rightarrow\quad \text{-}3a\,\ge\,\text{-}101 \quad\Rightarrow\quad a\,\le\,33$

Since $x \le 100\!:\;\;1-3a\,\le\,100 \quad\Rightarrow\quad \text{-}3a \,\le\,99 \quad\Rightarrow\quad a \,\ge\,\text{-}33$

. . Thus: . $a \in [\text{-}33,\,33]$

Then $a$ (and hence, $x$) can take on 67 values.

Therefore, there are $67$ lattice points on the line segment.