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The number of positive integers valued pairs (x, y) satisfying 4x -17y = 1 and x <= 1000 (x is less than or equal to 1000) is

(A) 59
(B) 57
(C) 55
(D) 58
(E) 60


x=(1+17y)/4......(1)
for x<1000 we have max value of y=235(a quick way for getting y=235 is just putting x=1000 in eq (1))

a note to make here that for (1+17Y) to be divisible by 4, Y should take only odd values

now if we look at pattern for different Y values for eq(1) to be divisible by 4
we find Y satisfying 3,7,11,15........
means for every 2 odds one is satisfying our condition.
So we need to find the half of total nos. of odds between our range 1 to 235(for x<1000)
We get 118 nos. of odd
and half of that is 118/2=59
Ans A.
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cleetus
The number of positive integers valued pairs (x, y) satisfying 4x -17y = 1 and x <= 1000 (x is less than or equal to 1000) is

(A) 59
(B) 57
(C) 55
(D) 58
(E) 60

We are asked to find integer solutions for x,y.

There is no need to apply Euclidian algorithm here, we can get particular solutions simply by plugging in some values.
As y reaches 3 we get:

y’ = 3 and x’ = 13. We need only to solve for x.

General solution for x will be:

x = x’ + bn = 13 + 17n

Hence lower limit for n is n ≥ 0.

Now we’ll find upper limit using given restriction x ≤ 1000. Plugging in general solution we’ll get:

13 + 17n ≤ 1000
17n ≤ 987
n ≤ 58

And our list of possible values of n has following representation:

0, 1,2, 3, ….. , 58

Total number of elements is 59.
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I have found an absolutely elegant way to solve this type of problems :

The number of positive integers valued pairs (x, y) satisfying 4x -17y = 1 and x <= 1000 (x is less than or equal to 1000) is :

1. Take the equation and divide by 4(the lesser of the two linear coeffs), giving :
x - 4.25y = .25

2. Take fractions to one side and integers to another, e.g. -
x - 4y = (y + 1)/4
and equate it with a constant k, e.g. -
x - 4y = (y + 1)/4 = k

3. From the above equation we can derive :
y = 4k - 1 ......................(1)
x = 4y + k......................(2)
replace (1) in (2)
x = 17k - 4 ....................(3)

From (1) we can derive k > 0, since y is positive.
From (2) we get 17k - 4 <= 1000
or, 17k <= 1004
or, k <= 1004/17
or, k <= 59....

So k can take values from 1 to 59, and each value of K gives a distinct pair of (x,y). Hence answer is 59. :):)
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