Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Including 1 and 60 itself, there are 12 distinct factors for 60.
The prime factors of 60 are 2, 3, 5, and its factor pairs are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10). We've put this below in a table for easy sharing.
| Factors of 60: | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 |
| Prime Factors of 60: | 2, 3, 5 |
| Factor Pairs of 60: | (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10) |
How to calculate factors?
To be a factor of 60, a number must divide 60 exactly, leaving no remainder. In other words, when 60 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 60 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 60 using four methods: basic factorization, prime factorization, the division method, or using GCD and LCM. We also include a detailed analysis of factor pairs and a factor tree to illustrate the breakdown.
Method 1: Basic Factorization
Basic Factorization is a method to find the factors of a number by systematically testing each whole number from 2 up to the number itself to see which ones divides with zero remainder (evenly). The process is somewhat time consuming if a number is high, that's why you should master divisibility rules, to make the process faster.
Here's the breakdown for 60:
| Divisor | Is it a factor of 60? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 60 = 60 |
| 2 | Yes, 60 is an even number so it's divisible by 2. | 2 × 30 = 60 |
| 3 | Yes, the sum of its digits (6) is divisible by 3. | 3 × 20 = 60 |
| 4 | Yes, the last two digits (60) form a number divisible by 4. | 4 × 15 = 60 |
| 5 | Yes, 60 ends with 0 or 5, so it's divisible by 5. | 5 × 12 = 60 |
| 6 | Yes, 60 is divisible by both 2 and 3. | 6 × 10 = 60 |
| 7 | No, 60 divided by 7 leaves a remainder of 4. | - |
| 8 | No, the last three digits (60) do not form a number divisible by 8. | - |
| 9 | No, the sum of its digits (6) is not divisible by 9. | - |
| 10 | Yes, 60 ends with 0, so it's divisible by 10. | 10 × 6 = 60 |
| 11 | No, the difference between sums of alternating digits (6) is not divisible by 11. | - |
| 12 | Yes, 60 is divisible by both 3 and 4. | 12 × 5 = 60 |
| ... | continue with all the other numbers. |
Method 2: Prime Factorization
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means a prime number cannot be formed by multiplying two smaller natural numbers. For example:
- 2 is a prime number because its only divisors are 1 and 2.
- 3 is prime for the same reason—it can only be divided evenly by 1 and 3.
- 4 is not prime because it can be divided by 1, 2, and 4.
- 5, 7, 11, and 13 are also prime numbers.
Prime numbers are fundamental in mathematics because they are the "building blocks" of whole numbers. Any natural number greater than 1 can be expressed as a product of prime numbers, which is known as its prime factorization.
How to do prime factorization of 60
You start by dividing 60 to each prime number, multiple times, until the remainder is 0. Then you move on to the next prime number. To save time, you should test with up to \( \sqrt60 \)
| Prime Number | Is it a factor of 60? | Verification |
|---|---|---|
| 2 | Yes, 60 is divisible by 2. | 60 ÷ 2 = 30, R0 |
| 2 | Yes, the result 30 is divisible by 2. | 30 ÷ 2 = 15, R0 |
| 2 | No, the result 15 is not divisible by 2. | - |
| 3 | Yes, 15 is divisible by 3. | 15 ÷ 3 = 5, R0 |
| 3 | No, the result 5 is not divisible by 3. | - |
| 5 | 5 is a prime number. | 5 is prime. |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 60 by performing successive divisions. This method involves dividing 60 by every integer from 1 up to 60 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 60, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 60 ÷ 1 = 60 |
| 2 | 60 ÷ 2 = 30 |
| 3 | 60 ÷ 3 = 20 |
| 4 | 60 ÷ 4 = 15 |
| 5 | 60 ÷ 5 = 12 |
| 6 | 60 ÷ 6 = 10 |
| 10 | 60 ÷ 10 = 6 |
| 12 | 60 ÷ 12 = 5 |
| 15 | 60 ÷ 15 = 4 |
| 20 | 60 ÷ 20 = 3 |
| 30 | 60 ÷ 30 = 2 |
| 60 | 60 ÷ 60 = 1 |
Using the division method, we calculated that factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Factor Tree of 60
The factor tree of 60 shows the step-by-step breakdown of 60 into its prime factors. Each branch of the tree represents a division of 60 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 60 and highlights the structure of its prime factorization.
| 60 | |||
| | | \ | ||
| 2 | 30 | ||
| | | \ | ||
| 2 | 15 | ||
| | | \ | ||
| 3 | 5 |
Factor Pairs of 60 (Visualization)
Factor pairs of 60 are sets of two numbers that, when multiplied together, result in 60. Factor pairs are symmetric and mirror around the square root of 60, such as (1, 60) and (60, 1), and can be both positive and negative pairs as long as their product equals 60.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -60) | (1, 60) |
| (-2, -30) | (2, 30) |
| (-3, -20) | (3, 20) |
| (-4, -15) | (4, 15) |
| (-5, -12) | (5, 12) |
| (-6, -10) | (6, 10) |
All factor pairs of 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10), (-1, -60), (-2, -30), (-3, -20), (-4, -15), (-5, -12), (-6, -10).
Why Should I Care About Factors of 60?
Turns out, factors aren’t just about boring math equations—they’re like secret superpowers hiding inside numbers! Knowing them can help you split things up, share with friends, or even spot hidden patterns. Want to know how? Check out these real-life examples that show just how cool factors really are:
- Hosting a Party: You have 60 party favors and want to give each guest the same amount. If there are 6 guests, each guest gets 10 party favors because 6 × 10 = 60.
- Arranging Flowers: You have 60 flowers to arrange. If each vase holds 5 flowers, you’ll need 12 vases because 5 × 12 = 60.
- Packing Snacks: You have 60 snack packs to prepare for a trip. If each person needs 1 snack packs, you’ll prepare enough for 60 people because 1 × 60 = 60.
- Stocking a Pantry: You have 60 cans of food to stock in the pantry. If each shelf holds 2 cans, you’ll need 30 shelves because 2 × 30 = 60.
- Packing a Picnic: You have 60 picnic items to pack. If each basket holds 1 items, you’ll need 60 baskets because 1 × 60 = 60.
