Factors & Prime Factorisation Calculator

Factors and Prime Numbers Guide

What's the key thing to understand about Finding All Factors?

A factor of n is any integer that divides n exactly (with no remainder). To find all factors: test every integer from 1 to √n. If i divides n, both i and n/i are factors. Example for 36: √36 = 6. Test 1 (36/1=36✓), 2 (36/2=18✓), 3 (36/3=12✓), 4 (36/4=9✓), 5 (no), 6 (36/6=6✓). Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. Note: 6 appears only once because 36/6=6.

What's the key thing to understand about Prime Factorisation?

Every integer greater than 1 can be written as a unique product of prime numbers — this is the Fundamental Theorem of Arithmetic. To find prime factorisation, divide by the smallest prime (2) repeatedly, then try 3, 5, 7, etc. Example: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2 × 2 × 2 × 3 × 15 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5. Uses: simplifying fractions, finding HCF and LCM, solving equations in number theory.

What do I need to know about Highly Composite Numbers?

Some numbers have unusually many factors because of their prime factorisation structure. 12 = 2² × 3 has 6 factors. 24 = 2³ × 3 has 8. 60 = 2² × 3 × 5 has 12. 360 = 2³ × 3² × 5 has 24 factors — this is why 360 was chosen for degrees in a circle: it is divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. Similarly 12, 24, and 60 for hours, months, and minutes.

What do I need to know about The Number of Factors Formula?

If n = p₁^a × p₂^b × p₃^c..., the number of factors = (a+1)(b+1)(c+1)... For 360 = 2³ × 3² × 5¹: (3+1)(2+1)(1+1) = 4×3×2 = 24 factors. This formula explains why square numbers always have an odd number of factors (because one factor pair has equal values), while all other numbers have an even count. A prime number always has exactly 2 factors: 1 and itself.

What do I need to know about HCF and LCM via Factors?

Once you have the full factor list for two numbers, the highest common factor (HCF) is simply the largest number appearing in both lists, and the lowest common multiple (LCM) follows from HCF × LCM = a × b. Example: 24 has factors 1, 2, 3, 4, 6, 8, 12, 24; 36 has factors 1, 2, 3, 4, 6, 9, 12, 18, 36. The shared factors are 1, 2, 3, 4, 6, 12 — so HCF = 12. Then LCM = (24 × 36) / 12 = 72. This factor-list method is slower than prime factorisation for large numbers but is the clearest way to see why HCF and LCM are related, and it's exactly what's taught at GCSE before prime factor trees are introduced. HCF is used to simplify fractions to lowest terms (divide numerator and denominator by their HCF) and LCM is used to find a common denominator when adding fractions with different denominators.

What do I need to know about Perfect, Abundant, and Deficient Numbers?

The sum of a number's proper divisors (all factors except the number itself) classifies it. A perfect number equals the sum of its proper divisors: 28's proper divisors are 1, 2, 4, 7, 14, which sum to exactly 28. Only four perfect numbers exist below 10,000 (6, 28, 496, 8128), all discovered by the ancient Greeks and all even — no odd perfect number has ever been found. An abundant number's proper divisors sum to more than the number: 12's proper divisors (1, 2, 3, 4, 6) sum to 16. A deficient number's proper divisors sum to less: 15's proper divisors (1, 3, 5) sum to only 9. Most numbers are deficient; prime numbers are always deficient since their only proper divisor is 1. This classification, dating to Euclid, has no major practical use today but remains a staple of number theory and recreational maths because it links so directly to the factors calculator's core output.

What do I need to know about Divisibility Rules as a Shortcut to Finding Factors?

Quick divisibility rules let you test potential factors without doing full division. A number is divisible by 2 if its last digit is even; by 3 if its digit sum is divisible by 3; by 4 if its last two digits form a number divisible by 4; by 5 if it ends in 0 or 5; by 6 if it passes both the rule for 2 and the rule for 3; by 9 if its digit sum is divisible by 9; and by 10 if it ends in 0. For example, 4,572: it's even (divisible by 2), its digit sum is 4+5+7+2=18 which is divisible by 9 (so also by 3), and its last two digits, 72, are divisible by 4 — confirming 2, 3, 4, and 9 are all factors without performing a single long division. These rules speed up manual factor-finding considerably before resorting to systematic division or a factor tree.

Worked Example: Using Prime Factorisation to Count Divisors

How many factors does 360 have? First find the prime factorisation: 360 = 2³ × 3² × 5¹. The number-of-factors formula adds 1 to each exponent and multiplies the results: (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24 factors. Listing them confirms it: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 — exactly 24 values. This formula is far faster than listing and counting by hand for larger numbers: 2,520 = 2³×3²×5×7 has (3+1)(2+1)(1+1)(1+1) = 4×3×2×2 = 48 factors, which would take much longer to verify by trial division alone.

Worked Example: Greatest Common Factor via Prime Factorisation

Find the greatest common factor (GCF, also called HCF) of 48 and 180 using prime factorisation. 48 = 2⁴ × 3¹. 180 = 2² × 3² × 5¹. The GCF takes the LOWEST power of each prime that appears in both factorisations: for 2, the lower power is 2² (since 48 has 2⁴ but 180 only has 2²); for 3, the lower power is 3¹ (48 has 3¹, 180 has 3²); the prime 5 doesn't appear in 48's factorisation at all, so it's excluded entirely. GCF = 2² × 3¹ = 4 × 3 = 12. Checking: 48/12 = 4 and 180/12 = 15, and since 4 and 15 share no common factors, 12 is confirmed as the greatest common factor. This method scales cleanly to any number of integers and any size of number, which is why prime factorisation — rather than listing every factor and comparing — is the standard method taught for finding GCF/HCF once numbers get larger than simple examples.

Worked Example: Checking Whether a Number Is Perfect

A perfect number equals the sum of its proper divisors (all factors except itself). Check 28: its factors are 1, 2, 4, 7, 14, 28. The proper divisors (excluding 28 itself) are 1, 2, 4, 7, 14, which sum to 1+2+4+7+14 = 28 — exactly matching the number itself, confirming 28 is perfect. Perfect numbers are rare: the first four are 6, 28, 496, and 8128, and all known perfect numbers are even (whether an odd perfect number can exist is still an open question in number theory). Compare this to 12, whose proper divisors (1, 2, 3, 4, 6) sum to 16 — greater than 12, making it 'abundant' rather than perfect, since its divisors collectively exceed the number itself.

Worked Example: LCM via Prime Factorisation

Find the least common multiple (LCM) of 18 and 24. Their prime factorisations are 18 = 2¹ × 3² and 24 = 2³ × 3¹. The LCM takes the HIGHEST power of each prime appearing in either number: for 2, the higher power is 2³ (from 24); for 3, the higher power is 3² (from 18). LCM = 2³ × 3² = 8 × 9 = 72. This is the mirror-image process to finding the GCF (which takes the lowest powers instead): GCF(18,24) = 2¹ × 3¹ = 6, and as a useful cross-check, GCF × LCM should always equal the product of the two original numbers: 6 × 72 = 432, and 18 × 24 = 432 ✓.