Factors & Divisor Counting
Choosing a divisor means independently choosing how many copies of each prime factor to include, so divisor counts arise from multiplying exponent-choice counts.
What you'll learn
- Construct every positive divisor from allowed prime exponents.
- Explain why exponent zero includes 1 and why full exponents include the number itself.
- Derive and use the divisor-count formula by multiplying independent choice counts.
- Apply exponent constraints to count selected kinds of divisors without duplication.
Before you start
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Worked recap: See why 72 has twelve positive divisors
Write 72 = 2³ × 3². Each positive divisor chooses an exponent for 2 from 0, 1, 2, 3 and an exponent for 3 from 0, 1, 2.
These independent choices give 4 × 3 = 12 divisors. Each exponent pair produces exactly one divisor.