Factors and Multiples: Formulas, Rules and Examples

Factors and multiples describe the divisibility relationships between numbers. This topic covers factor pairs, prime factorisation, the number of factors, common factors, common multiples, HCF, LCM and divisibility shortcuts. The formulas and examples help solve factors questions and multiples questions quickly and accurately.

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What Are Factors and Multiples?

A factor of a number divides it exactly without leaving a remainder. A multiple is obtained by multiplying a number by a whole number.

For a positive integer n, its positive factors are positive integers that divide n exactly. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. The multiples of 12 are 12, 24, 36, 48 and so on. Every positive number is a factor of itself, and every number is a multiple of 1.

Factors and Multiples Formula & Tricks

Important Formulas

Number of factors
If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then d(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1)

Here, p₁, p₂, ..., pₖ are distinct prime factors and a₁, a₂, ..., aₖ are their powers.

Sum of all factors
σ(n) = [(p₁ᵃ¹⁺¹ − 1)/(p₁ − 1)] × [(p₂ᵃ²⁺¹ − 1)/(p₂ − 1)] × ... × [(pₖᵃᵏ⁺¹ − 1)/(pₖ − 1)]

This formula gives the sum of all positive factors of n from its prime factorisation.

Product of all factors
Product of factors of n = n^(d(n)/2)

This follows because factors occur in pairs whose product is n. The formula also works when n is a perfect square.

HCF and LCM relation
For two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b

This relation is useful when any three of the four values are known.

Number of multiples
Multiples of n up to N = ⌊N/n⌋

The floor value gives the greatest whole number not exceeding N/n. It counts positive multiples n, 2n, 3n, and so on.

Quick Tricks

Find factors in pairs

For a number n, test divisors only up to √n. Whenever d divides n, record both d and n/d. If n is a perfect square, its square root is recorded only once.

Example: For 36, the pairs are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). Thus, 36 has 9 factors.
Use prime powers for factor count

First write the number as a product of prime powers. Add 1 to each exponent and multiply the results.

Example: 360 = 2³ × 3² × 5¹, so the number of factors is (3 + 1)(2 + 1)(1 + 1) = 24.
Use divisibility tests before division

Check simple rules for 2, 3, 5, 9, 10 and 11 before testing larger divisors. This reduces repeated calculations.

Example: 7,425 is divisible by 5 because it ends in 5, by 3 because 7 + 4 + 2 + 5 = 18, and by 9 because 18 is divisible by 9.

Factors and Multiples Concepts

Prime Factorisation and Number of Factors

The number of positive factors is found from the exponents in the prime factorisation.

If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then each factor can use 0 through aᵢ copies of pᵢ. Therefore, the total number of factors is (a₁ + 1)(a₂ + 1)...(aₖ + 1). A number with exactly two positive factors is prime.

Example: 72 = 2³ × 3². Hence, the number of positive factors of 72 is (3 + 1)(2 + 1) = 12.

Factor Pairs and Perfect Squares

Factors of a number can be arranged in pairs whose product equals that number.

For n, each divisor d has a corresponding divisor n/d. Non-square numbers have factor pairs with two different members. A perfect square has one unpaired factor, its square root, so its total number of factors is odd.

Example: 49 has the pairs (1, 49) and (7, 7), so it has 3 positive factors. Since 49 is a perfect square, its factor count is odd.

HCF and LCM Through Prime Factors

HCF uses the smallest powers of common prime factors, while LCM uses the greatest powers appearing in any number.

For 18 = 2 × 3² and 24 = 2³ × 3, HCF = 2¹ × 3¹ = 6, whereas LCM = 2³ × 3² = 72. The product relation gives 6 × 72 = 18 × 24.

Example: For 12 = 2² × 3 and 30 = 2 × 3 × 5, HCF = 2 × 3 = 6 and LCM = 2² × 3 × 5 = 60.

Divisibility Rules

A number is divisible by another number when the division leaves remainder zero.

A number is divisible by 2 if its last digit is even; by 3 if the sum of its digits is divisible by 3; by 4 if its last two digits are divisible by 4; by 5 if it ends in 0 or 5; by 8 if its last three digits are divisible by 8; by 9 if its digit sum is divisible by 9; and by 10 if it ends in 0. For 11, the difference between the sums of alternate digits must be 0 or a multiple of 11.

Example: 4,356 is divisible by 4 because 56 is divisible by 4, and it is divisible by 9 because 4 + 3 + 5 + 6 = 18.

Factors and Multiples Video Lessons

Watch short topic-wise lessons for quick revision.

14 Lessons
Lesson 1 of 14 Quick Revision

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.

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Practice Factors and Multiples Questions

Practise published questions related to this topic.

1What is the smallest natural number that should be added to 5023, such that a remainder of 5 is left when the resulting number is divided by each of the numbers 20, 98, 35 and 21?→ 258P8 is divisible by 9. What is the value of P?→ 3Find the square root of 64009.→ 4Determine the largest number that divides both 1058 and 1550, leaving remainders of 5 and 11, respectively.→ 5Find the smallest number which when divided by 9, 12 and 15 leaves a remainder 3 in each case.→ 616 m, 20 m and 24 m are the dimensions of a room. Find the length (in m) of the greatest possible scale to measure all dimensions exactly.→ 7Which of the following number is neither divisible by 9 nor by 11?→ 8√ 90.25 + √ 6,25 - √ 2.25 + √ 0.25 = ?→ 9If x = √225 + √484, what is the value of x2 - [(√225)2 + (32)2]?→ 10Three bells ring together at 3:00 PM. They ring at intervals of 18 minutes, 24 minutes, and 32 minutes. After how much time will they ring together again?→

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Quick Revision Notes

Factors and Multiples: Quick Revision

Use these rules and formulas for rapid revision.

  • A factor divides a number exactly; a multiple is obtained by multiplying a number by a whole number.
  • If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then d(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1).
  • A perfect square has an odd number of positive factors; every other positive integer has an even number.
  • To list factors, test divisors up to √n and record factor pairs.
  • HCF uses minimum prime exponents, while LCM uses maximum prime exponents.
  • For two positive integers, HCF × LCM = product of the two integers.
  • The number of positive multiples of n not exceeding N is ⌊N/n⌋.
  • A number is divisible by 3 or 9 when its digit sum is divisible by 3 or 9, respectively.

Factors and Multiples FAQs

How many positive factors does 540 have?

540 = 2² × 3³ × 5. Therefore, the number of factors is (2 + 1)(3 + 1)(1 + 1) = 24.

What is the difference between a factor and a multiple?

A factor divides a number exactly. A multiple is the result of multiplying a number by an integer; for example, 4 is a factor of 20, while 20 is a multiple of 4.

How many positive multiples of 7 are there up to 50?

The count is ⌊50/7⌋ = 7. They are 7, 14, 21, 28, 35, 42 and 49.

Why does a perfect square have an odd number of factors?

Factors usually occur in pairs, d and n/d. For a perfect square, the pair containing √n has equal members, so one factor remains unpaired.

What is the sum of the positive factors of 28?

The factors are 1, 2, 4, 7, 14 and 28. Their sum is 56. Using 28 = 2² × 7, σ(28) = (1 + 2 + 4)(1 + 7) = 7 × 8 = 56.

What are the HCF and LCM of 16 and 24?

16 = 2⁴ and 24 = 2³ × 3. HCF = 2³ = 8, and LCM = 2⁴ × 3 = 48.

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