Are you looking for a quick guide to all the factors and multiples of numbers from 1-100? If so, look here before carrying on!
Otherwise…
42 is an interesting number! Not only is it the answer to everything in the universe, according to The Hitchhiker’s Guide to the Galaxy, but it also has lots of nice mathematical properties.
We will break 42 down into its factors, look at the properties of these, then learn how to factorize any number.
Hang on a minute. What is a factor?
A factor is the same a divisor: a number that divides another number exactly, without leaving a remainder.
To think of it the other way round, factors are the numbers multiplied to make another number exactly.
42’s Factors
Pairs
Primes
Proper
Factorizing 42
Divisibility Rules
How to Find 42’s Factors
Prime Factorization of 42
42 is a Mathematically Special Number
Isn’t 42 Interesting?
To Sum Up (Pun Intended!)
All the Factors of 42
Let’s look at the number 42. It has 8 factors: 1, 2, 3, 6, 7, 14, 21, 42.
It has 6 proper factors: 2, 3, 6, 7, 14, 21. Their sum is 54.
We call a number abundant if it’s smaller than its sum of proper factors, so 42 is abundant.
In the rest of this lesson, we’re going to take an in-depth look at how we found this out and lots more about factors.
Factor Pairs of 42
From the definition, you can see that the factors of a positive whole number come in pairs.
The second divisor in each pair is found by dividing the original number by the pair’s first divisor. This ensures that our factors give the original number when multiplied together.
8 is a factor of 32. Its pair is (4, 8) and multiplying them gives:
The numbers in a pair are different, most of the time. This is not true for square numbers – their square root is a factor that is happy by itself!
Square numbers always have an odd number of factors because the same divisor isn’t counted twice. Numbers that aren’t squares have an even amount – they are all in pairs with numbers different to themselves.
50 has 6 factors: 1, 2, 5, 10, 25, 50
25 has 3 factors: 1, 5, 25
These are the pairs that multiply to make 42:
(21, 2)
(14, 3)
(7, 6)
If the rules were bent a little and included negative numbers, there would be another 8 divisors. We could make each one negative, and our equations would still make sense.
Remember:
Negative numbers multiply to make positive numbers, but positive multiplied by negative is negative! Both factors in each pair must have the same sign.
42 = -21 × -2
42 = -14 × -3
42 = -7 × -6
Prime Factors of 42
Prime factors, as you’ve probably guessed, are divisors AND prime numbers. Every number can be written as a product of its prime factors.
Breaking numbers down into products of primes allows us to write them more simply. It lets us look at the building blocks of a number before they are combined.
A prime number is one that can only be divided by 1 or itself.
So 42 has prime factors 2, 3, and 7.
We will look at the finding these in more detail below.
Proper Factors
Proper factors are simply the divisors of a number that aren’t 1 or the number itself. This means that a number’s proper factors are also in the category of its “regular” factors as above.
This table gives some examples of the difference between normal and proper factors.
| # | Factors | Proper Factors |
|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 2, 3, 4, 6 |
| 14 | 1, 2, 7, 14 | 2, 7 |
| 101 | 1, 101 | None! |
Do you notice the rule? Every number has two more factors than proper factors. Prime numbers like 101 only have two factors, so they always have 0 proper factors!
Factorizing 42
Now that we have our factors let’s work backward to see how we found them! We will also look at a harder example to give you the confidence to solve problems alone.
The simplest way to find the factors of a number is to use divisibility rules or a calculator. You might find it easier to start from 1 and work upwards.
Divisibility Rules
The divisibility rules up to 6 are as follows:
- All integers are divisible by 1
- All even integers are divisible by 2
- Add up the digits. If the sum is divisible by 3, the original number is
- Just look at the last two digits. If this number is divisible by 4, the original number is
- If the last digit is 0 or 5, the number is divisible by 5
- If the number is divisible by 2 and 3 using the above rules, it is also divisible by 6
Of course, if you have a calculator then you can simply divide 42 by each of the numbers to see if you get a whole number!
When you find a number that goes into 42 without a remainder, we write it down because it is a factor. We can find its pair by dividing 42 by the factor you’ve just found.
We stop our search once we reach the square root of 42, which is 6.48… We have already found all factors beyond this point – they are the pairs of the smaller factors!
How to Find the Factors of 42
42 is small, so you could probably find the factors in your head. However, it’s important to get the method right on small numbers before moving on to a bigger challenge!
1 is the first factor because all integers are divisible by 1.
1’s pair is (1, 42)
42 is even, so it is divisible by 2.
2’s pair is (2, 21)
Adding 42’s digits gives 4+2=6, and 6 is divisible by 3, so 42 is divisible by 3.
3’s pair is (3, 14)
Neither 4 nor 5 divides 42.
42 is divisible by 6 because it’s divisible by both 2 and 3
6’s pair is (6, 7)
We don’t need to check any for any factors higher than 6 because the square root of 42 is less than 7.
The list of factors of 42 is: 1, 2, 3, 6, 7, 14, 21, 42
Prime Factorization of 42
To find the prime factorization of 42, pick a pair of proper factors that include its smallest prime number – we don’t start with 1 and 42. If the prime’s matching number is not prime, factorize it.
For 42, the smallest prime factor is 2, so start by dividing 42 by 2. Then divide the result by its smallest prime factor.
We repeat this process until we cannot breakdown the factors anymore! It may help to think of this process visually as a tree.
The smallest prime factor of 42 is 2, and
2 is prime, so circle it and leave it alone.
However, 21 is divisible by numbers other than 1 and itself, so divide it by its smallest prime factor, 3:
3 and 7 are both prime, so we are done.
We substitute our factorization of 21 into our factorization of 42 to give a single equation:
This is the prime factorization of 42.
Your final two challenges for this lesson are to find all the factors of a much bigger number yourself.
Let’s look at a bigger number, 196. We will use the same method as before to find its factors.
1 is a factor. 1’s pair is (1, 196).
2 is a factor because 196 is even. 2’s pair is (2, 98).
3 is not a factor because 1+9+6=16, which is not divisible by 3.
4 is a factor because 96 is divisible by 4. 4’s pair is (4, 49).
5 is not a factor because the last digit is not 0 or 5.
6 is not a factor because 3 is not a factor.
7 is a factor. 7’s pair is (7, 28).
8 is not a factor because 196÷8=24.5 which is not an integer. The same is true for 9, 10, 11, 12, and 13.
14 is a factor. 14’s pair is 196÷14=14. We only need to write down 14 once.
We stop checking here because (14, 14), so any higher factors are already accounted for, in pairs with smaller factors.
The factors of 196 are: 1, 2, 4, 7, 14, 28, 49, 98, 196
To find the prime factorization of 196, pick the pair of proper factors that include the smallest prime number, 2.
98 is not prime, so you must factorize it. Find its smallest prime factor: it is divisible by 2.
49 is not prime however, it isn’t divisible by 2 so you must find the next smallest prime number that 49 can be divided by
Work your way through the prime numbers checking if they will divide 49.
3? No. 5? No. 7? Yes! Okay, so break 49 down into its factor pair that includes 7.
Finally, you’re left with only prime numbers. Like before, substitute the factorizations into the first equation:
To tidy things up, we collect repeated terms and express them as exponents. So the prime decomposition of 196 is:
= 72 × 22
Try this again, but this time choose a different pair to start with. Do you get the same answer?
42 is a Mathematically Special Number
As well as being a nice example for factoring, 42 has several special properties that are used in a branch of math called number theory:
- 42 is a Harshad number. This means that the sum of its digits is one of its factors: 4+2=6, and 6 is a factor since 6×7=42
- 42=2+4+6+8+10+12, so 42 is the sum of the first six even numbers
- 42 is a pronic number because it’s the product of two consecutive numbers, 6 and 7
- 42 is also called a sphenic number because it’s the product of three different primes. The first sphenic number is 30 which has prime factorization 30=5×2×3. The second is 42 which has prime factors 2, 3 and 7
Isn’t 42 Interesting?
That’s enough math for now! You might come across 42 again, though. Beyond the classroom, 42 is quite a famous number!
- In the famous fictional book The Hitchhiker’s Guide to the Galaxy, the answer to the “Ultimate Question of Life, the Universe, and Everything” which is calculated by Deep Thought, a supercomputer, after thinking for 7.5 million years, is 42! Unfortunately, your teacher won’t give many marks if you put 42 as the answer to everything in your next test!
- Times Square in New York City is on 42nd street
- Hagrid tells Harry Potter that he’s a wizard on page 42 of Harry Potter and the Philosopher’s Stone
To Sum Up (Pun Intended!)
Today, you have built up a strong foundation in factorization. We first defined “basic” factors, then thought about prime and proper factors too.
Prime factors are important because they are the basic building blocks of all integers – remember that every integer can be written uniquely as a product of prime factors!
We stated the factors of 42 and found a fool-proof method to find the factors of any number. Then, we used this technique to factorize 42 ourselves and find its prime factorization.
Factoring 196 took a little longer but used the same technique: Counting from 1, check if the number N is divisible, and find the factor’s pair if it is. Stop counting when you reach The square root of N.
We hope you’ve enjoyed today’s topic and feel confident in factoring numbers. Keep practicing, and leave a comment if you have any questions or a good fact about 42!
See our other factor lessons:
Factors of 56
Factors of 100