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

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Math Expert
Joined: 02 Sep 2009
Posts: 58366
A line passes through A(1,1) and B(100,1000). How many other points wi  [#permalink]

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22 Mar 2019, 01:20
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Difficulty:

65% (hard)

Question Stats:

34% (02:04) correct 66% (03:02) wrong based on 32 sessions

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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

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Math Expert
Joined: 02 Aug 2009
Posts: 7959
Re: A line passes through A(1,1) and B(100,1000). How many other points wi  [#permalink]

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22 Mar 2019, 20:20
Bunuel wrote:
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

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
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Re: A line passes through A(1,1) and B(100,1000). How many other points wi   [#permalink] 22 Mar 2019, 20:20
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