The number of distinct positive integral factors of 2014 is ___________.

GATE 2014 · Discrete Mathematics · Number Theory · easy

Answer: The number of distinct positive integral factors of 2014 is 8.

1. Prime factorisation of 2014: 2014 / 2 = 1007. Is 1007 prime? 1007 / 7 is not integer; 1007 / 11 is not; 1007 / 13 is not; 1007 / 17 is not; 1007 / 19 = 53. Is 53 prime? Yes (not divisible by 2, 3, 5, 7 and sqrt(53) < 8). So 2014 = 2^1 * 19^1 * 53^1.
2. Count divisors: tau(2014) = (1+1)(1+1)(1+1) = 2 * 2 * 2 = 8.