Factors, Multiples & Prime Factorisation
Factors and multiples are fundamental ideas of Number System and are closely connected with divisibility, HCF, LCM, prime numbers and many arithmetic problems. Prime factorisation expresses a number completely in terms of prime factors and provides a powerful base for solving competitive-exam questions quickly.
1. What is a Factor?
A positive integer a is called a factor or divisor of a positive integer n if n can be divided by a without leaving any remainder.
n = a × k
where k is also an integer.
Example: 6 is a factor of 24 because:
24 = 6 × 4
or equivalently, 24 ÷ 6 = 4 with remainder 0.
Convention used in this chapter: Unless otherwise stated, “factors” means positive factors, which is the usual convention in competitive arithmetic questions.
2. Factors of a Number
To find the factors of a number, identify all positive integers that divide it exactly.
Example: Factors of 12
1, 2, 3, 4, 6, 12
Each of these divides 12 without leaving a remainder.
3. Basic Properties of Factors
- 1 is a factor of every positive integer.
- Every positive integer is a factor of itself.
- A positive factor of n cannot be greater than n.
- A positive integer has only a finite number of positive factors.
- If a is a factor of n, then n is a multiple of a.
Important: For every positive integer n, both 1 and n are always factors of n.
4. Factor Pairs
Factors can often be arranged in pairs whose product equals the given number.
Example: Factor pairs of 36
- 1 × 36
- 2 × 18
- 3 × 12
- 4 × 9
- 6 × 6
Therefore, the positive factors of 36 are:
1, 2, 3, 4, 6, 9, 12, 18, 36
Exam Method: When listing factors manually, checking possible divisors only up to √n is enough to generate factor pairs. Once one factor is found, the corresponding paired factor is n divided by that factor.
5. Why Factor Pairs Meet Around √n
If n = a × b and a ≤ b, then a cannot be greater than √n. Otherwise both factors would exceed √n and their product would exceed n.
Therefore, while searching for factor pairs, we need to test possible smaller factors only up to √n.
Example: √36 = 6, and the final factor pair is 6 × 6.
6. Factors of a Prime Number
A prime number has exactly two distinct positive factors:
1 and the number itself
Examples:
- Factors of 7: 1, 7
- Factors of 13: 1, 13
- Factors of 29: 1, 29
7. Factors of a Composite Number
A composite number has more than two distinct positive factors.
Example: Factors of 18 are:
1, 2, 3, 6, 9, 18
Therefore, 18 is composite.
8. The Special Case of 1
The number 1 has exactly one positive factor:
1
That is why 1 is neither prime nor composite.
9. What is a Multiple?
A number obtained by multiplying an integer by another integer is called its multiple.
If:
m = n × k
for some integer k, then m is a multiple of n.
Example: 35 is a multiple of 7 because:
35 = 7 × 5
10. Positive Multiples of a Number
The positive multiples of n are:
n, 2n, 3n, 4n, 5n, ...
Example: Positive multiples of 6 are:
6, 12, 18, 24, 30, 36, ...
11. Multiples are Infinite
A positive integer has infinitely many positive multiples because it can be multiplied by 1, 2, 3, 4, ... without end.
Remember: Factors of a positive integer are finite, but its positive multiples are infinite.
12. Is Zero a Multiple?
Yes. Under the integer definition of multiple:
0 = n × 0
Therefore, 0 is a multiple of every non-zero integer.
Exam Convention: If a question asks for the “first positive multiples” of a number, start with n, 2n, 3n, ... and do not include 0.
13. Basic Properties of Multiples
- Every positive integer is a positive multiple of itself.
- A positive integer has infinitely many positive multiples.
- Every positive multiple of n is greater than or equal to n.
- If m is a multiple of n, then n is a factor of m.
- The sum of two multiples of n is also a multiple of n.
- The difference of two multiples of n is also a multiple of n.
14. Factor and Multiple are Reverse Relations
If a divides b exactly, then:
a is a factor of b ⇔ b is a multiple of a
Example:
Since 8 divides 40 exactly:
- 8 is a factor of 40.
- 40 is a multiple of 8.
Exam Shortcut: “a is a factor of b” and “b is a multiple of a” express the same divisibility relationship from opposite directions.
15. Factor vs Multiple — Quick Comparison
| Property | Factor | Multiple |
|---|
| Meaning | Divides the number exactly | Obtained by multiplying the number by an integer |
| For a positive integer n | Finite in number | Positive multiples are infinite |
| Size | Positive factor ≤ n | Positive multiple ≥ n |
| Example for 12 | 1, 2, 3, 4, 6, 12 | 12, 24, 36, 48, ... |
16. Common Factor
A number that is a factor of each of two or more given numbers is called a common factor.
Example:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Therefore, the common factors are:
1, 2, 3, 6
The greatest common factor leads to the concept of HCF/GCD, which is studied separately in Chapter 3.
17. Common Multiple
A number that is a multiple of each of two or more given numbers is called a common multiple.
Example:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Some common multiples are:
12, 24, 36, 48, ...
The least positive common multiple leads to the concept of LCM, which is studied separately in Chapter 3.
18. Prime Factor
A factor of a number that is itself prime is called a prime factor.
Example: The positive factors of 60 include 2, 3 and 5, and all three are prime.
Therefore, the distinct prime factors of 60 are:
2, 3 and 5
19. What is Prime Factorisation?
Prime factorisation means expressing an integer greater than 1 as a product of prime numbers only.
Example:
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
This is the prime factorisation of 60.
Important: A prime factorisation must contain only prime factors. An expression such as 60 = 6 × 10 is a factorisation, but it is not yet a prime factorisation because 6 and 10 are composite.
20. Fundamental Theorem of Arithmetic
Every integer greater than 1 is either a prime number itself or can be expressed as a product of prime numbers in a unique way, apart from the order of the factors.
Example:
84 = 22 × 3 × 7
The order may be changed, but the prime factors and their exponents remain the same.
Key Idea: The prime factorisation of an integer greater than 1 is unique except for the order in which the prime factors are written.
21. Prime Factorisation by Repeated Division
In the repeated-division method, divide the number successively by the smallest possible prime divisor until the quotient becomes 1.
Example: Prime factorisation of 360
| Prime Divisor | Quotient |
|---|
| 2 | 360 ÷ 2 = 180 |
| 2 | 180 ÷ 2 = 90 |
| 2 | 90 ÷ 2 = 45 |
| 3 | 45 ÷ 3 = 15 |
| 3 | 15 ÷ 3 = 5 |
| 5 | 5 ÷ 5 = 1 |
Therefore:
360 = 23 × 32 × 5
22. Prime Factorisation by Factor Tree
A number may also be repeatedly split into smaller factors until every terminal factor is prime.
Example: Factor tree idea for 72
72 = 8 × 9
8 = 2 × 2 × 2
9 = 3 × 3
Therefore:
72 = 23 × 32
Important: Different valid factor trees always lead to the same final prime factorisation.
23. Writing Prime Factorisation in Exponential Form
Repeated prime factors should generally be written using powers.
Example:
540 = 2 × 2 × 3 × 3 × 3 × 5
Therefore:
540 = 22 × 33 × 5
This compact form is especially useful in later questions involving HCF, LCM, factors, perfect squares and perfect cubes.
24. Distinct Prime Factors vs Repeated Prime Factors
It is important to distinguish between distinct prime factors and the total occurrence of prime factors.
Example:
72 = 23 × 32
The distinct prime factors are:
2 and 3
Although 2 occurs three times and 3 occurs twice in the expanded prime factorisation.
Exam Trap: If a question asks for the “distinct prime factors,” count each prime only once. If it asks for prime factors with multiplicity, repeated occurrences matter.
25. Smallest and Greatest Prime Factors
Once a number has been written in prime-factor form, its smallest and greatest prime factors can be identified immediately.
Example:
1260 = 22 × 32 × 5 × 7
Therefore:
- Smallest prime factor = 2
- Greatest prime factor = 7
Quick Check: For any even integer greater than 2, the smallest prime factor is always 2.
26. Prime Factorisation of a Prime Number
If a number is already prime, its prime factorisation is simply the number itself.
Examples:
- Prime factorisation of 17 = 17
- Prime factorisation of 29 = 29
- Prime factorisation of 101 = 101
27. Does 1 Have a Prime Factorisation?
No. The number 1 has no prime factors.
The Fundamental Theorem of Arithmetic applies to integers greater than 1.
Important Exam Fact: 1 has no prime factor and no prime factorisation.
28. Solved Example: Factor or Multiple?
Consider the numbers 9 and 63.
Since:
63 = 9 × 7
we can immediately conclude:
- 9 is a factor of 63.
- 63 is a multiple of 9.
29. Solved Example: Find All Factors Using Factor Pairs
Find all positive factors of 48.
Factor pairs are:
- 1 × 48
- 2 × 24
- 3 × 16
- 4 × 12
- 6 × 8
Therefore, the factors are:
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
30. Solved Example: Prime Factorisation
Find the prime factorisation of 420.
420 = 42 × 10 = (2 × 3 × 7)(2 × 5)
Therefore:
420 = 22 × 3 × 5 × 7
Exam Approach: Whenever possible, break the number into familiar factors and continue until every factor is prime.
31. If a is a Factor of b
If a is a factor of b, then b can be written as:
b = ak
for some integer k. This also means that b is a multiple of a.
Equivalent statements: a divides b ⇔ a is a factor of b ⇔ b is a multiple of a.
32. Transitive Property of Factors
If a is a factor of b and b is a factor of c, then a is also a factor of c.
If b = ax and c = by, then:
c = axy
Therefore, a divides c.
Example: 3 divides 12 and 12 divides 60, so 3 divides 60.
33. Every Factor of a Factor is Also a Factor
If d is a factor of a, and a is itself a factor of n, then d must also be a factor of n.
Example: 6 is a factor of 30, and 3 is a factor of 6. Therefore, 3 is also a factor of 30.
34. Multiples of a Multiple
If b is a multiple of a and c is a multiple of b, then c is also a multiple of a.
Example: 24 is a multiple of 6 and 120 is a multiple of 24. Therefore, 120 is also a multiple of 6.
35. Sum of Multiples
If x and y are both multiples of n, then x + y is also a multiple of n.
Let x = an and y = bn. Then:
x + y = (a + b)n
Example: 35 and 70 are multiples of 5, so 35 + 70 = 105 is also a multiple of 5.
36. Difference of Multiples
If x and y are multiples of n, then x − y is also a multiple of n.
x − y = (a − b)n
Example: 84 and 36 are multiples of 12, and 84 − 36 = 48 is also a multiple of 12.
37. Integer Multiple of a Multiple
If x is a multiple of n, then every integer multiple of x is also a multiple of n.
If x = kn, then for any integer m:
mx = mkn
Therefore, n divides mx.
38. Linear Combination Property
If a number d divides both x and y, then d also divides every integer linear combination of x and y:
mx + ny
where m and n are integers.
Example: Since 5 divides both 20 and 35, it also divides 3 × 20 + 2 × 35 = 130.
This property becomes especially useful later in HCF and remainder problems.
39. Prime Factorisation of a Product
To prime-factorise a product, combine the prime factors of its factors.
Example:
36 × 50 = (22 × 32)(2 × 52)
Therefore:
36 × 50 = 23 × 32 × 52
40. Prime Factorisation of a Power
If:
n = paqbrc
then for a positive integer k:
nk = pakqbkrck
Example: 12 = 22 × 3, so:
123 = 26 × 33
41. Divisibility Through Prime Exponents
Suppose two positive integers have prime-factor forms:
A = paqbrc... and B = pxqyrz...
A divides B only when every prime appearing in A occurs in B with at least the same exponent.
Prime-Exponent Test: A | B if, for every prime factor of A, its exponent in A is less than or equal to its exponent in B.
42. Example of Prime-Exponent Divisibility
Determine whether 72 divides 1080.
72 = 23 × 32
1080 = 23 × 33 × 5
1080 contains at least three factors of 2 and at least two factors of 3. Therefore:
72 divides 1080
43. Example Where Divisibility Fails
Does 48 divide 360?
48 = 24 × 3
360 = 23 × 32 × 5
48 requires four factors of 2, but 360 contains only three.
Therefore, 48 does not divide 360.
44. Prime Factors of a Divisor
If a positive integer a divides another positive integer b, then every prime factor of a must also be a prime factor of b.
Example: Since 45 divides 315:
45 = 32 × 5
both 3 and 5 must occur in the prime factorisation of 315.
Important: The converse requires sufficient exponents. Merely containing the same distinct prime factors does not guarantee divisibility.
45. Same Prime Factors, Different Numbers
Two numbers can have exactly the same distinct prime factors but still be different because their exponents may differ.
Example:
12 = 22 × 3
18 = 2 × 32
Both have distinct prime factors 2 and 3, but they are different numbers.
46. Number Made from Only One Prime Factor
A positive integer greater than 1 may have only one distinct prime factor. Such a number has the form:
pk
where p is prime and k ≥ 1.
Examples: 8 = 23, 27 = 33, 125 = 53.
Only when k = 1 is the number itself prime; for k > 1 it is composite.
47. Product of Distinct Prime Numbers
If a number is the product of two or more prime numbers, it is composite.
Examples:
- 15 = 3 × 5
- 35 = 5 × 7
- 66 = 2 × 3 × 11
Each has factors other than 1 and itself.
48. Product of Two Distinct Primes
If p and q are distinct primes, then pq has the positive factors:
1, p, q, pq
Example: 35 = 5 × 7 has factors 1, 5, 7 and 35.
The detailed relationship between prime exponents and the number of factors will be studied systematically in Topic 1.11.
49. Prime Square
If p is prime, then p2 has only the distinct prime factor p.
Example:
121 = 112
Its only distinct prime factor is 11.
50. Recognising a Perfect Square from Prime Factorisation
A positive integer is a perfect square if and only if every exponent in its prime factorisation is even.
Example:
900 = 22 × 32 × 52
All exponents are even, so 900 is a perfect square.
Perfect-square methods are studied in greater depth in Chapter 5 : Square & Square Root.
51. Recognising a Perfect Cube from Prime Factorisation
A positive integer is a perfect cube if and only if every exponent in its prime factorisation is divisible by 3.
Example:
1728 = 26 × 33
Both exponents are divisible by 3. Therefore, 1728 is a perfect cube.
Perfect-cube concepts will be studied in detail in Chapter 6 : Cube & Cube Root.
52. Prime Factorisation Helps Compare Large Numbers
Prime factorisation can reveal relationships that are less obvious in ordinary form.
Example:
216 = 23 × 33
540 = 22 × 33 × 5
From the factorisations, we can immediately compare their prime factors and determine which factors are shared and which are not.
53. Finding a Missing Factor
If:
N = a × b
and N and a are known, the other factor is:
b = N/a
Example: If 924 = 28 × k, then:
k = 924/28 = 33
54. Solved Example: Missing Prime Factor
Suppose:
630 = 2 × 32 × 5 × p
Find p.
Known product:
2 × 9 × 5 = 90
Therefore:
p = 630/90 = 7
Hence the missing prime factor is 7.
55. Solved Example: Which Number is a Factor?
Which of 6, 8, 10 and 14 is a factor of 126?
126 = 2 × 32 × 7
- 6 = 2 × 3 → divides 126.
- 8 requires 23 → does not divide 126.
- 10 requires a factor 5 → does not divide 126.
- 14 = 2 × 7 → also divides 126.
Therefore, both 6 and 14 are factors of 126.
Question-design lesson: In an MCQ asking for exactly one answer, the options must be chosen so that only one satisfies the condition.
56. Solved Example: Prime Factorisation of a Large Number
Find the prime factorisation of 2772.
2772 ÷ 2 = 1386
1386 ÷ 2 = 693
693 ÷ 3 = 231
231 ÷ 3 = 77
77 = 7 × 11
Therefore:
2772 = 22 × 32 × 7 × 11
57. Solved Example: Check Divisibility Using Prime Factors
Does 180 divide 3780?
180 = 22 × 32 × 5
3780 = 22 × 33 × 5 × 7
Every prime exponent required by 180 is available in 3780. Therefore:
180 divides 3780
58. Solved Example: Common Prime Factors
Find the common prime factors of 84 and 150.
84 = 22 × 3 × 7
150 = 2 × 3 × 52
The prime factors appearing in both numbers are:
2 and 3
Therefore, 2 and 3 are their common prime factors.
59. Common Exam Traps
Trap 1: A factor and a multiple are not the same thing. If 4 is a factor of 20, then 20 is a multiple of 4.
Trap 2: Positive factors of a positive integer are finite, whereas its positive multiples are infinite.
Trap 3: 1 is a factor of every positive integer, but 1 has no prime factorisation.
Trap 4: 0 is a multiple of every non-zero integer, but when a question asks for positive multiples, 0 is excluded.
Trap 5: A factorisation is not necessarily a prime factorisation. Every final factor must be prime.
Trap 6: Different factor trees may look different, but they must give the same final prime factorisation.
Trap 7: “Distinct prime factors” means repeated copies of the same prime are counted only once.
Trap 8: Sharing the same distinct prime factors does not mean two numbers are equal; their exponents may differ.
Trap 9: For A to divide B, B must contain every prime factor of A with at least the required exponent.
Trap 10: Do not confuse listing factors with calculating their number, sum or product; those formulas are handled separately in Topic 1.11.
60. Quick Revision
- A factor divides a number exactly without leaving a remainder.
- 1 and the number itself are always positive factors of every positive integer.
- Positive factors of n cannot exceed n.
- Positive factors of a positive integer are finite.
- A multiple is obtained by multiplying a number by an integer.
- Positive multiples of n are n, 2n, 3n, 4n, ...
- A positive integer has infinitely many positive multiples.
- 0 is a multiple of every non-zero integer.
- a is a factor of b if and only if b is a multiple of a.
- Factors can be generated efficiently through factor pairs up to √n.
- A prime has exactly two distinct positive factors.
- A composite number has more than two distinct positive factors.
- 1 has exactly one positive factor and is neither prime nor composite.
- A common factor divides each of the given numbers.
- A common multiple is a multiple of each of the given numbers.
- A prime factor is a factor that is itself prime.
- Prime factorisation expresses a number greater than 1 entirely as a product of primes.
- Prime factorisation is unique apart from the order of the prime factors.
- 1 has no prime factorisation.
- Repeated division and factor trees give the same final prime factorisation.
- Repeated prime factors are conveniently written using exponents.
- If a divides b and b divides c, then a divides c.
- If d divides both x and y, then d divides mx + ny for all integers m and n.
- If A divides B, every prime factor of A occurs in B with at least the required exponent.
- The prime factorisation of a product is obtained by combining prime exponents.
- Raising a number to a positive integral power multiplies all its prime exponents by that power.
- A perfect square has even exponents in its prime factorisation.
- A perfect cube has prime exponents divisible by 3.
- Distinct prime factors are counted without repetition.
Practice these verified previous-year questions based on factors, multiples and prime factorisation. These questions also show how prime factorisation is applied in common-multiple and perfect-square problems.
Practice these exam-oriented questions on factors, multiples, factor pairs, prime factors, prime factorisation and prime-exponent properties. Try each question before opening the answer and explanation.