Counting Divisors — Total Factors of 120 How many positive factors (divisors) does 120 have in total?

Introduction / Context:
Counting divisors relies on prime factorization and a standard formula. This is a staple of number system questions and tests whether you can decompose a number quickly and apply the multiplicative rule for divisor counts.

Given Data / Assumptions:

• Target number: 120.
• We seek the count of all positive divisors, including 1 and the number itself.

Concept / Approach:
Factor 120 into primes. If N = p1^a * p2^b * p3^c, then the total number of divisors d(N) = (a+1)(b+1)(c+1). This counts choices for each prime power exponent from 0 up to its maximum.

Step-by-Step Solution:
Prime factorization: 120 = 2^3 * 3^1 * 5^1.Apply formula: d(120) = (3+1)(1+1)(1+1) = 4 * 2 * 2.Compute: 4 * 2 * 2 = 16.Therefore, 120 has 16 positive divisors.

Verification / Alternative check:
List quickly in pairs (1×120, 2×60, 3×40, 4×30, 5×24, 6×20, 8×15, 10×12) to see eight pairs → 16 divisors.

Why Other Options Are Wrong:
3 and 10 are far below the correct count; 12 misses several divisors; 18 exceeds the true total.

Common Pitfalls:
Incorrect prime factorization (e.g., missing a factor 3); forgetting to add 1 to each exponent when applying the formula.

Final Answer:
16