if positive integers m and n satisfy 23/50<m/n<64/139,find the smallest possible value of n. ans:63, plz explain?

23/50 < m/n < 64/139
23n < 50m and 64n > 139m
Let 50m - 23n = x ≥ 1 ... ① and 64n - 139m = y ≥ 1 ... ②
①×139 + ②×50 :
3200n - 3197n = 139x + 50y
n = (139x + 50y)/3 ≥ (139(1) + 50(1))/3 = 63 while m = (23(63)+1)/50 = 29 ,
we have 23/50 < 29/63 < 64/139 , the smallest possible value of n is 63.