Let f be the set of all fractions m/n where m and n are positive integers with m + n s 2005. If a be the largest number in f such that a < 16/23 expressed in lowest form as m/n. Find (n - m)/6.

Question

Please solve it

Answer 1

m/n=m/(2005-m) because m+n=2005.

m/n<16/23, m/(2005-m)<16/23,

23m<16(2005-m), 23m<32080-16m, 39m<32080, m<32080/39.

32080/39=822.56 approx., therefore m≤822.

When m=822, n=1183, m/n=822/1183<16/23 and is the largest fraction in f which is less than 16/23.

(n-m)/6=(1183-822)=361/6.