Factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. Including 1 and 42 itself, there are 8 distinct factors for 42.
The prime factors of 42 are 2, 3, 7, and its factor pairs are (1, 42), (2, 21), (3, 14), (6, 7). We've put this below in a table for easy sharing.
| Factors of 42: | 1, 2, 3, 6, 7, 14, 21, 42 |
| Prime Factors of 42: | 2, 3, 7 |
| Factor Pairs of 42: | (1, 42), (2, 21), (3, 14), (6, 7) |
How to calculate factors?
To be a factor of 42, a number must divide 42 exactly, leaving no remainder. In other words, when 42 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 42 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 42 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 42:
| Divisor | Is it a factor of 42? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 42 = 42 |
| 2 | Yes, 42 is an even number so it's divisible by 2. | 2 × 21 = 42 |
| 3 | Yes, the sum of its digits (6) is divisible by 3. | 3 × 14 = 42 |
| 4 | No, the last two digits (42) do not form a number divisible by 4. | - |
| 5 | No, last digit is 2, so not divisible by 5. | - |
| 6 | Yes, 42 is divisible by both 2 and 3. | 6 × 7 = 42 |
| 7 | Yes, 42 divided by 7 equals 6 with no remainder. | 7 × 6 = 42 |
| 8 | No, the last three digits (42) do not form a number divisible by 8. | - |
| 9 | No, the sum of its digits (6) is not divisible by 9. | - |
| 10 | No, last digit is 2, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (2) is not divisible by 11. | - |
| 12 | No, 42 is not divisible by both 3 and 4. | - |
| ... | 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 42
You start by dividing 42 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 \( \sqrt42 \)
| Prime Number | Is it a factor of 42? | Verification |
|---|---|---|
| 2 | Yes, 42 is divisible by 2. | 42 ÷ 2 = 21, R0 |
| 2 | No, the result 21 is not divisible by 2. | - |
| 3 | Yes, 21 is divisible by 3. | 21 ÷ 3 = 7, R0 |
| 3 | No, the result 7 is not divisible by 3. | - |
| 7 | 7 is a prime number. | 7 is prime. |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 42 by performing successive divisions. This method involves dividing 42 by every integer from 1 up to 42 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 42, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 42 ÷ 1 = 42 |
| 2 | 42 ÷ 2 = 21 |
| 3 | 42 ÷ 3 = 14 |
| 6 | 42 ÷ 6 = 7 |
| 7 | 42 ÷ 7 = 6 |
| 14 | 42 ÷ 14 = 3 |
| 21 | 42 ÷ 21 = 2 |
| 42 | 42 ÷ 42 = 1 |
Using the division method, we calculated that factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
Factor Tree of 42
The factor tree of 42 shows the step-by-step breakdown of 42 into its prime factors. Each branch of the tree represents a division of 42 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 42 and highlights the structure of its prime factorization.
| 42 | ||
| | | \ | |
| 2 | 21 | |
| | | \ | |
| 3 | 7 |
Factor Pairs of 42 (Visualization)
Factor pairs of 42 are sets of two numbers that, when multiplied together, result in 42. Factor pairs are symmetric and mirror around the square root of 42, such as (1, 42) and (42, 1), and can be both positive and negative pairs as long as their product equals 42.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -42) | (1, 42) |
| (-2, -21) | (2, 21) |
| (-3, -14) | (3, 14) |
| (-6, -7) | (6, 7) |
All factor pairs of 42 are (1, 42), (2, 21), (3, 14), (6, 7), (-1, -42), (-2, -21), (-3, -14), (-6, -7).
Why Should I Care About Factors of 42?
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:
- Filling Water Bottles: You have 42 liters of water and want to fill water bottles. If each bottle holds 3 liters, you will fill 14 bottles because 3 × 14 = 42.
- Cooking Dinner: You have 42 pieces of chicken to cook. If each person gets 1 pieces, you’ll need to serve 42 people because 1 × 42 = 42.
- Building Stairs: You have 42 steps to build for a staircase. If each section includes 2 steps, you’ll have 21 sections because 2 × 21 = 42.
- Making Cookies: You have 42 cookie dough balls to bake. If each tray holds 1 dough balls, you’ll need 42 trays because 1 × 42 = 42.
- Stacking Books: You have 42 books to stack in a library. If each stack has 1 books, you’ll have 42 stacks because 1 × 42 = 42.
