A line passes through A(1,1) and B(100,1000). How many other points wi

Question

A line passes through A(1,1) and B(100,1000). How many other points with integer coordinates are on the line and strictly between A and B?

(A) 0
(B) 2
(C) 3
(D) 8
(E) 9

chetan2u · Accepted Answer

The line is (1,1) to (100,1000), so the slope of line m is  \frac{1000-1}{100-1}=\frac{999}{99}=\frac{111}{11} 
Thus the slope form of the equation can be written as .. y=mx....y= \frac{111}{11} x
Thus, for all points on line, the slope should be same, so if we take a point (x,y), the slope from (x,y) to (1,1) should be  \frac{111}{11} 
So,  \frac{y-1}{x-1}=\frac{111}{11}.....11y-11=111x-111......11y=111x-100.....11y=11x+100x-100......11(y-x)=100(x-1) 
  Since x and y are integers and 11 and 100 are co-prime, x-1 should be a multiple of 11...  
Thus x-1 can be 0,11,22,33,44,55,66,77,88,99. The corresponding x values will be 1,12,23,34,45,56,67,78,89,100, so total 10 values.
But 1 and 100 are already the end points of the line, and we are looking at the points in between, so 10-2=8 values

D

TheNightKing · Answer

chetan2u

"Jahapana tussi great ho!! Tofa kabul karo" - Bollywood Movie "3 Idiots"

Sir, you're great. I don't know how you solve all these questions flawlessly.

Nicely done. Thank you!

Fdambro294 · Answer

Step 1: find the equation of the line satisfying the 2 points

Slope = m = (1,000 - 1) / (100 - 1) = 999/99 = 111/11

Y = (111/11) X + B

Plugging in the coordinates (1 , 1) which fall on the line we find that the

The Y intercept = B = (-)(100/11)

Equation of given line is:

Y = (111/11)X - (100/11)

—multiply both sides by 11 and you have the Linear Indeterminate Equation of:

11Y + 100 = 111X

We are asked for the Integer Coordinate Pairs of (X , Y) that fall in between the 2 given points (1 , 1) and (100 , 1,000)

Step 2: we know that when X = 1 and Y = 1 the equation of: 11y + 100 = 111x : will be satisfied

Concept: because the variables appear on each side of the equation, the Corresponding Integer Pairs that Satisfy the Equation will follow this pattern

When you have a linear indeterminate equation such as:

(1) 
a(X) + b(Y) = c

Where a and b and c = constant number values and X and Y = unknown Integer Values

(1st) find the first integer X, Y pair that satisfies the equation. Then each consecutive integer pair that satisfies the equation can be found by:

X + b (add the coefficient in front of the Y unknown variable)

And

Y - (a) (subtract the coefficient in front of the X unknown variable)

(2)
for a linear indeterminate equation of the form:

a(X) + c = b(Y)

find each consecutive pair of Integer (X , Y) values that satisfy the equation by:

X + b (add the coefficient in front of the Y variable)

And

Y + a (add the coefficient in front of the X variable)

For:

11y + 100 = 111x

First integer pair is (1 , 1)

X = 1 ———> each consecutive X Integer Value that satisfies the equation will increase by the Coefficient in front of the Y variable —-> (+11)

Y = 1 ———-> each consecutive corresponding Y Integer value that satisfies the equation will increase by the Coefficient in front of the X variable (+111)

Thus, the next pair of Integer coordinate values that will satisfy the equation:

Start: (X = 1) and (Y = 1)

(X= 12) and (Y = 112)

Then: add (+11) to the X value again —-and —- and add (+111) to the Y value again

(X = 23) and (Y = 123)

....

This will continue until

(X = 100) and (Y = 1,000)

We can use the X integer values to determine how many Pairs of (X, Y) integer values satisfy the equation:

We need to count the integer values in between  , NON- inclusive, when the Increment or spacing between each consecutive value is (+11)

Non-inclusive Count =   - 1

= (99/11) - 1

= 9 - 1

8

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MathRevolution · Answer

All points between A and B will have the same slope as that of the given points.

Slope:  \frac{1000 - 1 }{ 100 - 1} = \frac{999 }{ 99} = \frac{111}{ 11} . This is  \frac{100 + 11 }{ 11} .

All points that will have the same slope must be expressed in terms of  111/11

=>  \frac{888}{88} , \frac{777}{77} , \frac{666}{66} , \frac{555}{55}, \frac{ 444}{44}, \frac{333}{33}, \frac{222}{22}, \frac{111}{11}

Eight points

Answer D

Crytiocanalyst · Answer

The points on the line are from (1,1) to (100,1000=111/11
Slope must be same for all 111/11
Since x and y are integers and they can all take the values 
888/88,777/77,666/66,555/55,444/44,333/33,222/22,111/11
total 8 values.
Hence IMO D

bumpbot · Answer

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A line passes through A(1,1) and B(100,1000). How many other points wi

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