List every factor of a number, along with its prime factors and how many factors it has.
Every number from 1 up to the square root is tested for division without remainder.
Each factor found gives a partner factor on the other side of the square root.
Prime factorisation repeatedly divides out the smallest prime that fits, until nothing is left.
Factor: A whole number that divides n with no remainder
Search range: Test only up to √n; factors come in pairs
Finding the factors of 360.
Number: 360
Divide out 2s: 360 = 2 × 2 × 2 × 45
Divide out 3s: 45 = 3 × 3 × 5
Prime factorisation: 2³ × 3² × 5
Factor count: (3+1) × (2+1) × (1+1) = 24
Answer: 360 has 24 factors; 2³ × 3² × 5
A factors calculator lists every whole number that divides a given number exactly, along with its prime factorisation and the count and sum of its factors. For 120, it shows all 16 factors from 1 to 120 and the factorisation 2³ × 3 × 5.
Factors are the starting point for simplifying fractions, finding HCF and LCM, and solving arrangement problems. How many ways can 120 chairs be set out in equal rows? Which box sizes pack a quantity with none left over? The factor list answers these at a glance. Teachers use it to prepare worksheets, and students use it to check factor pairs, perfect numbers and questions about the number of divisors.
1. Start with 1 and n, which are always factors.
2. Test every whole number k from 2 up to √n.
3. Whenever k divides n exactly, record both k and its partner n ÷ k.
4. If k × k = n, record k once only.
5. For the prime factorisation, divide repeatedly by the smallest prime that fits until 1 is left.
6. Check the count: if n = p₁^a × p₂^b × ..., the number of factors is (a + 1)(b + 1)....
Every factor k of n has a partner n ÷ k, and if k is below √n its partner is above it. So a search up to √n finds every pair. For 120, √120 is about 10.95, and the factors up to 10 are 1, 2, 3, 4, 5, 6, 8 and 10, with partners 120, 60, 40, 30, 24, 20, 15 and 12. When n is a perfect square, the square root pairs with itself, which is why perfect squares have an odd number of factors.
Any factor of n = 2³ × 3 × 5 must be built from the same primes with powers no larger than those in n. The power of 2 can be 0, 1, 2 or 3, four choices. The power of 3 can be 0 or 1, and so can the power of 5, two choices each. Each combination gives a different factor, so there are 4 × 2 × 2 = 16. This counting rule works for any number once you know its prime factorisation.
Factors divide into n; multiples are what n divides into. A number has finitely many factors but infinitely many multiples. The sum of factors is used to classify numbers: if the sum of factors other than n itself equals n, the number is perfect, as with 6 and 28. If it exceeds n, the number is abundant, and 120 is a clear example. These ideas appear in olympiad questions rather than board exams, but they rest on the same factor list.
Raghav is arranging 120 chairs for a school function and wants to see every way they can be set out in equal rows.
Divide by 2: 120 ÷ 2
Divide by 2: 60 ÷ 2
Divide by 2: 30 ÷ 2
Divide by 3: 15 ÷ 3
Remaining value is prime: 5 is prime — stop
Answer: Number of factors 16; All factors 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120; Prime factorisation 2^3 × 3 × 5
Missing the larger partner of each small factor, such as listing 8 but forgetting 15.
Counting the square root twice for a perfect square.
Confusing factors with multiples in word problems.
Writing prime factorisations that include composite numbers, such as 120 = 4 × 30.
Finding all equal-row arrangements for seating, planting or parade formations.
Choosing box or packet sizes that divide a quantity with nothing left over.
Finding HCF by comparing the factor lists of two numbers.
Simplifying fractions and square roots by spotting large factors.
Olympiad questions on number of divisors and perfect or abundant numbers.
How can I tell how many factors a number has without listing them?
Take the prime factors, add 1 to each power, and multiply them together. For 360 = 2³ × 3² × 5¹, that gives 4 × 3 × 2 = 24 factors.
Why does a perfect square have an odd number of factors?
Factors normally come in pairs, giving an even total. But in a perfect square, the square root pairs with itself, so it is only counted once instead of twice, making the total an odd number.