The number of distinct positive integral factors of 2014 is…

Step 1: Prime factorize 2014.

2014 = 2 × 1007 = 2 × 19 × 53, so the prime factorization is 2¹ × 19¹ × 53¹.

Step 2: Use the divisor-count formula. If n = pa x qb x rc …, the number of positive divisors is (a+1)(b+1)(c+1)…

Here the exponents are 1, 1, and 1, so the number of positive divisors is (1+1)(1+1)(1+1) = 2 × 2 × 2 = 8.

All positive divisors are:

• 1

• 2

• 19

• 38

• 53

• 106

• 1007

• 2014

Therefore, the number of distinct positive integral factors of 2014 is 8.