Even Factors: Formula, Rules and Solved Examples

Even Factors are positive factors of a number that are divisible by 2. Their count can be found directly from the prime factorisation of the number. For a number written as 2^a multiplied by odd prime powers, the number of even factors is obtained by choosing a positive exponent for 2 and any allowed exponent for each odd prime.

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What Are Even Factors?

Even factors are the positive divisors of a number that are even, meaning they are divisible by 2. If the number contains a factor of 2 in its prime factorisation, its even factors can be counted using the exponent of 2.

Let N = 2^a × p^b × q^c, where p and q are distinct odd primes. An even factor must contain 2^1, 2^2, ..., or 2^a, giving a choices for the power of 2. The powers of p and q can range from 0 to b and 0 to c, giving (b + 1)(c + 1) choices. Therefore, the number of even factors is a(b + 1)(c + 1). For example, 60 = 2^2 × 3 × 5, so its number of even factors is 2 × 2 × 2 = 8.

Even Factors Formula & Tricks

Important Formulas

General even factors formula
If N = 2^a × p₁^b¹ × p₂^b² × ... × pᵣ^bᵣ, then number of even factors = a × (b₁ + 1)(b₂ + 1)...(bᵣ + 1)

The exponent of 2 in an even factor can be any integer from 1 to a, giving a choices. Each odd prime exponent can independently take all values from 0 to its exponent in N.

Using total and odd factors
Number of even factors = Total number of factors − Number of odd factors

For N = 2^a × product of odd prime powers, total factors are (a + 1) multiplied by the odd-prime choices, while odd factors use exponent 0 for 2.

Odd number case
If N is odd, number of even factors = 0

An odd number has no factor divisible by 2, so it cannot have an even factor.

Quick Tricks

Subtract odd factors from total factors

When the complete factor count is easier to calculate, count the odd factors separately and subtract them from the total.

Example: For 72 = 2^3 × 3^2, total factors = 4 × 3 = 12 and odd factors = 3. Hence, even factors = 12 − 3 = 9.
Check the exponent of 2 first

If the number is odd, stop immediately because the answer is zero. If the exponent of 2 is a, it contributes exactly a choices, not a + 1, because exponent 0 would produce an odd factor.

Example: For 180 = 2^2 × 3^2 × 5, the factor 2 has 2 choices: 2^1 or 2^2. Thus, even factors = 2 × 3 × 2 = 12.

Even Factors Concepts

Counting Even Factors from Prime Factorisation

For N = 2^a × p₁^b¹ × p₂^b² × ..., the number of even factors is a × (b₁ + 1)(b₂ + 1)... .

An even factor must include at least one 2. Therefore, the exponent of 2 has a choices: 1 through a. For every odd prime, the exponent may be any value from 0 through its exponent in N. Multiplying these independent choices gives the count.

Example: For 360 = 2^3 × 3^2 × 5, the number of even factors is 3 × 3 × 2 = 18.

Even Factors by Total Minus Odd Factors

The number of even factors equals the total number of factors minus the number of odd factors.

For N = 2^a × p₁^b¹ × ..., total factors are (a + 1)(b₁ + 1)..., while odd factors are (b₁ + 1)... because the power of 2 must be zero. Their difference is a(b₁ + 1)... .

Example: For 96 = 2^5 × 3, total factors = 6 × 2 = 12 and odd factors = 2. Hence, the number of even factors is 12 − 2 = 10.

Even Factors of Odd and Even Numbers

An odd number has zero even factors, while every positive even number has at least one even factor.

If the prime factorisation has no factor 2, the number is odd and no divisor can be divisible by 2. If the exponent of 2 is at least 1, the number itself and suitable divisors containing 2 are even.

Example: The number 45 = 3^2 × 5 has 0 even factors. The number 14 = 2 × 7 has 1 even factor, namely 2.

Listing Even Factors

To list even factors, first list the factors of N that contain at least one factor of 2.

For a small number, factor pairs can be checked directly. For larger numbers, use the prime factorisation and select the exponent of 2 from 1 through a, while selecting every allowed exponent for the odd primes.

Example: For 24 = 2^3 × 3, the even factors are 2, 4, 6, 8, 12 and 24. Their count is 3 × 2 = 6.

Even Factors Video Lessons

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14 Lessons
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Practice Even Factors Questions

Practise published questions related to this topic.

1Three numbers are in the ratio 3 : 2 : 13, and their LCM is 2574. Their HCF is:→ 2What is the Least Common Multiple of 4/7, 8/14 and 12/21?→ 3Three numbers are in the ratio 3 : 7 : 2, and their LCM is 5376. Their HCF is:→ 4What is the number of ordered pairs (x, y) where x, y are positive integers such that the eight digit number 74x8935y is divisible by 11?→ 5A drone technician is programming a flight controller. He notices that three-fifths of the square of a certain safety distance value used in the code equals 126.15. He needs to determine the original safety distance to update the firmware correctly. What is that number?→ 6A is smallest 3 digit number divisible by both 12 and 15. B is the largest 3 digit number divisible by 12 and 15. What is the value of B - A?→ 7√(0.0225/0.04) = ?→ 8Find the smallest number which when increased by 20 is completely divisible by 12, 20, 15, 18 and 24.→ 9If the ratio of two numbers is 4 : 1, and their HCF is 47, then their LCM is:→ 10A data analyst is studying a series of zoomed-in satellite resolutions. The base square root reference she uses is: √18225 = 135 She now wants to quickly compute the total of square roots for four scaled-down values (by factors of 10): (√182.25 + √1.8225 + √0.018225 + √0.00018225) What is the value of this sum?→

Even Factors Quick Quiz

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Quick Revision Notes

Even Factors Quick Revision

Use the prime factorisation to count even divisors without listing them individually.

  • An even factor is a positive divisor divisible by 2.
  • If N is odd, its number of even factors is 0.
  • For N = 2^a × p₁^b¹ × p₂^b² × ..., even factors = a × (b₁ + 1)(b₂ + 1)... .
  • The exponent of 2 has a choices, from 1 through a; exponent 0 is excluded.
  • Even factors = total factors − odd factors.
  • For N = 2^a × odd prime powers, total factors use (a + 1), but even factors use a.

Even Factors FAQs

What is the formula for the number of even factors?

If N = 2^a × p₁^b¹ × p₂^b² × ..., then the number of even factors is a × (b₁ + 1)(b₂ + 1)... .

How many even factors does 120 have?

120 = 2^3 × 3 × 5. Therefore, the number of even factors is 3 × 2 × 2 = 12.

How many even factors does an odd number have?

An odd number has zero even factors because none of its divisors is divisible by 2.

What is the difference between total factors and even factors?

Total factors include both odd and even divisors. Even factors include only divisors containing at least one factor of 2, so their count is total factors minus odd factors.

How many even factors does 2^4 × 3^2 have?

The exponent of 2 gives 4 choices, and the exponent of 3 gives 3 choices. Thus, the number of even factors is 4 × 3 = 12.

How many even factors does 48 have?

48 = 2^4 × 3. The number of even factors is 4 × 2 = 8. They are 2, 4, 6, 8, 12, 16, 24 and 48.

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