Factors of 60: All 12 Factors, Prime Factorization and Factor Pairs

Q: How many factors does 60 have?

60 has exactly 12 positive factors. This makes 60 a highly composite number, the smallest number with 12 divisors.

Q: What is the greatest common factor of 60 and another number?

The greatest common factor (GCF) of 60 and another number is the largest factor they share. For example GCF(60, 48) is 12. To find it, list the factors of each number or use the prime factorization method.

Q: Why does 60 have so many factors?

60 is the product of three small primes: 2 squared times 3 times 5. Numbers built from many small primes have many divisors, which is why ancient civilizations (and modern timekeeping) chose 60 for minutes, seconds and arc degrees.

60 has 12 positive factors. They are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. This guide shows how to find them, why 60 has so many, the prime factorization, the factor tree, factor pairs, and the math behind GCF and LCM calculations that use the factors of 60.

Factors of 60

123 456 101215 203060

Sum of all 12 factors: 168. Prime factorization: 2 x 2 x 3 x 5 (also written 2² x 3 x 5).

What Is a Factor?

A factor of a whole number n is any whole number that divides n exactly, with no remainder. Factors always come in pairs because if a is a factor of n, then n divided by a is also a factor of n.

If a x b = 60, then both a and b are factors of 60.

Factor Pairs of 60 (Drawing)

Every factor of 60 has a partner. The 12 factors form 6 pairs that multiply to 60.

All 6 factor pairs of 60. Reading the small numbers gives 1, 2, 3, 4, 5, 6; the big numbers are 10, 12, 15, 20, 30, 60.

How to Find the Factors of 60 (Step by Step)

Start at 1

1 is a factor of every positive integer. Pair it with the number itself, 60. That gives the first pair: 1 x 60.

Test 2

60 is even, so 2 divides it. 60 / 2 = 30. Second pair: 2 x 30.

Test 3

Add the digits of 60: 6 + 0 = 6, which is divisible by 3, so 60 is divisible by 3. 60 / 3 = 20. Third pair: 3 x 20.

Test 4

60 / 4 = 15. Fourth pair: 4 x 15.

Test 5

60 ends in 0, so it is divisible by 5. 60 / 5 = 12. Fifth pair: 5 x 12.

Test 6

60 / 6 = 10. Sixth pair: 6 x 10.

Stop when the partner is smaller than the test number

Next test would be 7, but 60 / 7 is not a whole number. 8 and 9 are not factors either. The next would be 10, but 10 already showed up paired with 6. You have all the factors.

Final list: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Twelve factors total.

Prime Factorization of 60

The prime factorization expresses 60 as a product of prime numbers only.

60 = 2 x 2 x 3 x 5 = 2² x 3 x 5

This is the unique fingerprint of 60 in the integers, guaranteed by the fundamental theorem of arithmetic. Every positive integer above 1 has exactly one prime factorization (up to the order of the factors).

Factor Tree of 60

The factor tree is a visual way to derive the prime factorization. Split 60 into a pair, then split each composite number until everything is prime.

Factor tree starting from 60 -> 6 and 10, then splitting both composite numbers into their primes.

Why 60 Has So Many Factors

60 is built from the three smallest prime numbers: 2, 3 and 5. Numbers assembled from many small primes have many divisors. In math terms, the number of divisors of n is calculated from the exponents in its prime factorization:

If n = p₁^a x p₂^b x p₃^c, then d(n) = (a+1)(b+1)(c+1)

For 60 = 2² x 3¹ x 5¹, this gives (2+1)(1+1)(1+1) = 3 x 2 x 2 = 12 factors. That matches the list we found by hand.

60 is in fact a highly composite number, the smallest number with 12 divisors. The Babylonians chose base 60 for astronomy, and we still use it for minutes, seconds and angle degrees because of how easily 60 splits into halves, thirds, quarters, fifths, sixths and tenths.

Complete Factor Table

Factor	Partner	Product	Type
1	60	1 x 60 = 60	Identity
2	30	2 x 30 = 60	Prime / Composite
3	20	3 x 20 = 60	Prime / Composite
4	15	4 x 15 = 60	Composite / Composite
5	12	5 x 12 = 60	Prime / Composite
6	10	6 x 10 = 60	Composite / Composite

The prime factors of 60 are 2, 3 and 5. The composite factors are 4, 6, 10, 12, 15, 20, 30 and 60. The number 1 is neither prime nor composite by definition.

Sum, Count and Other Properties

Number of factors12 positive factors (or 24 if you count negative factors too).

Sum of factors1+2+3+4+5+6+10+12+15+20+30+60 = 168. This is sigma(60) in number theory.

Sum of proper factors168 – 60 = 108. Proper factors exclude the number itself.

Abundancy108 > 60, so 60 is an abundant number.

Highly composite60 is the smallest number with 12 divisors.

Common multiples60 is the LCM of {1, 2, 3, 4, 5, 6}, which is why a 60-minute hour divides evenly into many fractions.

Using the Factors of 60 in GCF and LCM

Greatest common factor (GCF)

To find GCF(60, X), list the factors of both numbers and take the largest one shared by both.

GCF(60, 48): factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Shared with 60: 1, 2, 3, 4, 6, 12. Largest: 12.
GCF(60, 36): shared factors are 1, 2, 3, 4, 6, 12. Largest: 12.
GCF(60, 25): shared factors are 1, 5. Largest: 5.

Least common multiple (LCM)

For LCM(60, X), use the prime factorizations and take the highest power of every prime that appears.

LCM(60, 48): 60 = 2² x 3 x 5, 48 = 2⁴ x 3. Highest powers: 2⁴, 3, 5. LCM = 16 x 3 x 5 = 240.
LCM(60, 25): 25 = 5². Highest powers: 2², 3, 5². LCM = 4 x 3 x 25 = 300.

Why 60 Is Everywhere in Daily Life

Time60 seconds in a minute, 60 minutes in an hour. The Babylonian sexagesimal system survived 4000 years for a reason.

Geometry360 degrees in a full circle = 60 x 6. Easy to divide a circle into halves, thirds, fourths, fifths, sixths and tenths.

MusicBPM tempos use 60 as a reference (60 BPM is one beat per second).

Currency systemsOld British coinage used shillings with 12 pence each; 5 shillings was 60 pence. Easy fractions for retail.

Negative Factors of 60

For most school problems “factor of 60” means a positive factor. If you include negatives, every positive factor has a negative partner because (-a) x (-b) also equals a x b. So 60 has 24 integer factors total:

±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60

Authoritative References

Wolfram MathWorld on divisors and the divisor function: MathWorld: Divisor
OEIS sequence A005179 (smallest number with n divisors): OEIS A005179
OEIS sequence A002182 (highly composite numbers, includes 60): OEIS A002182

FAQs

What are the factors of 60?

60 has 12 positive factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. Each divides 60 with no remainder.

What is the prime factorization of 60?

The prime factorization of 60 is 2 x 2 x 3 x 5, also written 2² x 3 x 5.

How many factors does 60 have?

60 has exactly 12 positive factors. This makes 60 the smallest highly composite number with 12 divisors.

What is the greatest common factor of 60 and another number?

The greatest common factor (GCF) is the largest factor shared by both numbers. For example GCF(60, 48) is 12 and GCF(60, 25) is 5.

Why does 60 have so many factors?

60 is built from the three smallest primes (2, 3 and 5), and one of them (2) appears squared. By the divisor formula, that gives (2+1)(1+1)(1+1) = 12 divisors. Many small prime factors means many divisors.

Is 60 a perfect number?

No. A perfect number equals the sum of its proper divisors. 60’s proper divisors sum to 108, more than 60 itself, so 60 is an abundant number.

Final Answer

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. The prime factorization is 2² x 3 x 5, and the formula for the number of divisors confirms there are exactly 12. Among small numbers, 60 has one of the richest factor sets, which is why so many measurement systems were built around it.

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