Odd Factors: Formula, Rules and Solved Examples
Odd Factors are divisors that are not divisible by 2. This page explains how to find the number of odd factors using prime factorisation, how powers of 2 affect the count, and how to solve odd divisor questions with direct formulas and examples.
What Are Odd Factors?
The factor 2 does not contribute to the count of odd factors because an odd factor contains no power of 2. First remove the complete power of 2 from n; the remaining odd part contains all the odd factors. For example, 72 = 2³ × 3². Its odd factors are the factors of 3², namely 1, 3 and 9, so it has 3 odd factors.
Odd Factors Formula & Tricks
Important Formulas
Ignore the exponent of 2 and multiply one more than each exponent of the odd prime factors.
The number of odd factors of n equals the total number of factors of its odd part.
The total factor count includes choices for the power of 2, whereas odd factors use only the zero power of 2.
Quick Tricks
For counting odd factors, remove every factor of 2 and factor only the remaining odd number.
Divide the number repeatedly by 2 until an odd number remains, then find the total factors of that odd number.
A positive odd number always has at least one odd factor, namely 1. Listing factors can verify the formula for small values.
Odd Factors Concepts
Prime Factorisation Method
If n = 2^a × p₁^b¹ × p₂^b² × ..., every odd divisor has the form p₁^c¹ × p₂^c² × ..., where 0 ≤ cᵢ ≤ bᵢ. Each exponent has bᵢ + 1 choices, giving the product formula.
Odd Part and Its Divisors
If n = 2^a × m and m is odd, then every odd divisor of n divides m. Thus, finding odd factors can be reduced to finding all factors of m. For n = 200, 200 = 2³ × 25, so its odd factors are the factors of 25: 1, 5 and 25.
Finding the Number of Odd Factors
The exponent of 2 is excluded. For n = 2^a × 3^b × 5^c, the count is (b + 1)(c + 1), regardless of the value of a.
Odd Factors of a Perfect Square
For an odd number m = p₁^2r¹ × p₂^2r² × ..., the odd-factor count is (2r₁ + 1)(2r₂ + 1)..., which is odd. The power of 2 in the original square does not change this conclusion.
Odd Factors Video Lessons
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Multiples and Factors in Number System
Understand the meaning of multiples and factors, learn how they are related, and identify them using simple examples in the Number System.
Practice Odd Factors Questions
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Odd Factors: Quick Revision
Use these rules to count odd divisors quickly and accurately.
- An odd factor is a positive divisor not divisible by 2.
- Write n = 2^a × product of powers of odd primes.
- The exponent of 2 is ignored when counting odd factors.
- Number of odd factors = product of one more than the exponents of the odd prime factors.
- The odd factors of n are the divisors of n after removing its highest power of 2.
- For n = 2^a × m with m odd, number of odd factors = d(m).
- A perfect square has an odd number of odd factors.
Odd Factors FAQs
What is the formula for the number of odd factors of a number?
If n = 2^a × p₁^b¹ × p₂^b² × ..., then the number of odd factors is (b₁ + 1)(b₂ + 1)...; the exponent of 2 is ignored.
How many odd factors does 360 have?
360 = 2³ × 3² × 5. Therefore, the number of odd factors is (2 + 1)(1 + 1) = 6.
How do you find the odd factors of 96?
96 = 2⁵ × 3. Its odd part is 3, whose factors are 1 and 3. Therefore, 96 has 2 odd factors.
Are all factors of an odd number odd?
Yes. An odd number cannot have an even factor, because multiplying an even factor by any integer gives an even number. Hence, all positive factors of an odd number are odd.
What is the difference between total factors and odd factors?
For n = 2^a × m, where m is odd, total factors are (a + 1)d(m), while odd factors are only d(m). The choices for powers of 2 are excluded from odd factors.
How many odd factors does 2^7 × 3^4 × 5² have?
Ignore 2⁷ and use the odd-prime exponents: (4 + 1)(2 + 1) = 15 odd factors.