The number of positive integers valued pairs (x, y) ssatisfying 4x -17y = 1 and x <= 1000 (x is less than or equal to 1000) is Choose the correct answer
A) fifty nine	B) 57	C) 55	D) 58	E) 60
4x-17y=1   y=(4x-1)/17   x has to be a positive integer such that 1&lt;=x&lt;=1000(4x-1) is a factor of 17 so that y is also a positive integer   So we need to know how many numbers in the set {3,7,11,15,....,3999} are divisible by 17   Lets try to observe the pattern   1. 3 mod 17 = 3   2. 7 mod 17 = 7   3. 11 mod 17 = 11   4. 15 mod 17 = 15   5. 19 mod 17 = 2   6. 23 mod 17 = 6   7. 27 mod 17 = 10   8. 31 mod 17 = 14   9. 35 mod 17 = 1   10. 39 mod 17 = 5   11. 43 mod 17 = 9   12. 47 mod 17 = 13   13.51 mod 17 = 0   14. 55 mod 17 = 4   15. 59 mod 17 = 8   16. 63 mod 17 = 12   17. 67 mod 17 = 16   ... and after this the remainders will repeat (this is no co-incidence that the cyclicity is also 17)   So the right choices are x=13,13+17,13+17+17,.....   The number of such numbers is 59 (highest one being 999) fifty nine
<result>A</result>
The number of positive integers valued pairs (x, y) ssatisfying 4x -17y = 1 and x <= 1000 (x is less than or equal to 1000) is