What is a factor of 100? How do I find them? Is a divisor different from a factor?

All of these questions will be answered in this lesson.

**A factor is a number that divides, or goes into, another number exactly.**

Another way of saying this is that the other number is a multiple of the factor.

In a fair world, sharing 12 chocolate bars between 4 people leaves each person with 3 chocolate bars.

In this case: **3 and 4 are factors of 12** because they can each divide 12 without any remainders – no leftover chunks of chocolate!

Divisor is **another word** for factor. Both words will be used interchangeably through this lesson so you can get more comfortable with them.

100’s Factors

Pairs

Primes

How to Find 100’s Factors

Divisibility Tests

Prime Factoriziation of 100

Isn’t 100 Interesting?

100 is a VERY Imporant Number

Why is 100 Everywhere?

To Sum Up (Pun Intended!)

## All the Factors of 100

The factors of 100 are **1, 2, 4, 5, 10, 20, 25, 50, and 100**.

If you want a quick reference, look at this useful table of the factors of 1 to 100, including primes, and the first 20 multiples.

Otherwise, read on and conquer the factor!

### Factor Pairs of 100

The pairs for 100 are each made from two whole numbers, or **integers**, that multiply together to make 100.

Since they’re integers, they can be positive or negative. Here they are in their pairs:

Positive Pair | Negative Pair |
---|---|

(1, 100) | (-1, -100) |

(2, 50) | (-2, -50) |

(4, 25) | (-4, -25) |

(5, 20) | (-5, -20) |

(10, 10) | (-10, -10) |

It doesn’t matter which order the multiplication number pairs are given in, because multiplication is commutative.

**Commutative** means number order can be swapped without making a difference to the answer.

Both

and

are equal to 100.

So factor pairs can be written either way round like this **(4, 25)** or like this **(25, 4)**.

They are both the same pair.

We’ll have a look at how we found these pairs later on.

### Prime Factors of 100

A prime number can be divided only by 1 and itself.

3 cannot be divided by another number apart from itself.

and

Its only factors are 1 and 3, so it is a prime number.

So from a number’s divisors, at least one MUST be a prime.

For now, 100’s positive factors are **1, 2, 4, 5, 10, 25, 50, and 100**

Of these numbers, *only 2 and 5 are prime*, so those are our prime factors.

## How to Find the Factors of 100

If you would rather read than watch, continue here.

A sensible way to find the factors of any number is to find the pairs, by starting with 1, 2, 3, etc., and checking if they divide the number that we want to factorize.

This means that when we find one divisor, we can divide our number by it – which finds another divisor!

That’s how you make a pair.

Remember…

All factors, apart from square roots, come in pairs like this. For a square number, when dividing it by its square root, we get the same number. For example **100 ÷ 10 = 10**.

Fortunately, there are some tricks that allow us to quickly check whether a number is divisible by another.

They’re called divisibility tests.

The table below shows some useful divisibility tests for the numbers 1 to 10, many of which you’ll already know, and whether they give a positive result applied to 100.

### Divisibility Tests Up to 10

Click the documents below for free, printable PDFs.

The first shows clearly the divisibility rules up to 15 in a neat table, great for checking your method as you go:

The second has 14 practice questions with answers, one question for each divisibility rule from 2 to 15:

Whether you use the worksheets above or not, below are the handy hints for checking if 100 can be divided by numbers from 1 to 10.

Divisible by …? | Test | Works for 100? |
---|---|---|

1 | No test needed – all numbers are divisible by 1 | ✅ |

2 | Even – number ends in 2, 4, 6, 8, or 0 | ✅ |

3 | Sum of digits is a multiple of 3 | ❌ |

4 | The number made by the last 2 digits are divisible by 4 | ✅ |

5 | Number ends in 0 or 5 | ✅ |

6 | Divisible by 2 and 3 | ❌ |

7 | No simple test! | ❌ |

8 | Divisible by 4 after being halved | ❌ |

9 | Sum of digits is a multiple of 9 | ❌ |

10 | Number ends in 0 | ✅ |

Check which ones work for 100 to get the smaller factors: 1, 2, 4, 5, and 10. Then divide 100 by each of these numbers – you’ll find the larger, matching part of the pair: 100, 50, 25, and 20.

This is a general method that works for all numbers, but for the small numbers, you need to check all the way up to the square root of your number.

Coming up you will see, there *is* a test for divisibility by 7, but it’s not quite as immediate as some of the others.

### Divisibility Test for 7

It is still useful to know, so here’s how it works.

- Subtract twice the last digit from the rest of the number
- Continue until the number is less than 70 or easy to divide by 7
- If the result is divisible by 7, then so was the original

Here’s an example. See if you can work out the answers before expanding the boxes to check!

= 35

= 7 × 5

So 392 is divisible by 7.

Okay, how about a much bigger number?

Use the same method as above, just apply it a few times.

= 644

644 ⟶ 64 -4 × 2

= 56

= 8 × 7

So 6,608 is also divisible by 7.

We’re fortunate that the square root of 100 is 10 because, for numbers bigger than 10, it gets a bit trickier to test for divisibility.

- Start with the digit on the left
- Subtract the digit to its right
- Then add the one after that
- Then subtract the one after that
- And so on
- If the resulting sum is a multiple of 11, then so is the original number.

Start by following 11’s divisibility rule, so alternately subtracting and adding one digit to the next, starting from the right.

8 – 1 + 9 – 5 = 11

= 1 × 11

So 8195 is a multiple of 11.

5 – 4 + 3 – 2 = 2

2 is not divisible by 11, so 5432 is not a multiple of 11.

- Add 4 times the last digit to the remaining digits
- If the result is divisible by 13, then the original number is too

You can use this process repeatedly to deal with larger numbers.

For example, let’s see if 2717 is divisible by 13:

= 299

= 65

= 13 × 5

So, 2717 is divisible by 13.

34 is not a multiple of 13, so neither is 3049.

Have a go finding your own divisibility tests for 12, 14, and 15. Then you check for divisibility all the way up to 15!

Hint: They are not complicated – look at the test for 6 and 8 for some ideas. Let us know what you came up with – comment below!

### Prime Factorization of 100

Returning to the idea from earlier, there is a unique way of writing 100 as a product of only its prime factors. This is called its **prime factorization** or **prime decomposition**.

100 can be split into any two of its divisors, then each of those numbers can be split up again into two more. This is repeated until you’re only left with primes which we can’t split up any further.

What you end up with is a **factor tree**, which looks like this:

This tree is shown by the equation:

=2

^{2}× 5

^{2}

So, the primes that divide 100 are 2 and 5, and 100 has the **unique** prime factorization 2^{2} × 5^{2}.

All integers have the special property that their prime decomposition is unique – there’s no other way to make 100 using only prime numbers!

We can use the prime decomposition of a number to find all its divisors.

Since any divisor of 100 is made up of some of its prime factors, we just remove some and see what we’re left with:

4 = 2 × 2

5 = 5

10 = 2 × 5

20 = 2 × 2 × 5

25 = 5 × 5

50 = 2 × 5 × 5

100 = 2 × 2 × 5 × 5

What if you don’t care about what the divisors are, just how many there are?

There are 9 different divisors of 100, including 1 and 100.

Remember…

Using the factors of 100 as an example, have another look at its prime decomposition:

^{2}× 5

^{2}

The number of times each prime appears can’t be more than its power. There can be no more than two 5’s or two 2’s in any divisor of 100.

But are you allowed to have zero of a particular prime? The answer is yes, so for each prime divisor, you have 3 choices for how many times it is used: 0, 1, or 2 times.

Since there are two primes for 100, you have **3 ^{2} = 3 × 3 = 9** choices in total.

Now for 1000. Have a go at using the factor tree method above for finding the prime factorization.

You should get **1000 = 2 ^{3} × 5^{3}**.

This means there are 4 choices of how many times each prime appears: 0, 1, 2, or 3 times. Use this fact to show that 1000 has

factors in total.

For 1500, the prime factorization is

^{2}× 3 × 5

^{3}

Meaning there are:

- 3 choices for the number of 2’s
- 2 choices for the number of 3’s
- 4 choices for the number of 5’s

The number of choices is the exponent + 1 to include zero as an option.

Multiplying the number of options together gives the number of possible combinations, so the number of divisors is:

There are 24 factors of 1500.

When you’ve finished with that, pick your favorite number and try to find its prime factorization. Then, see if you can work out how many factors it has!

And now that you know how to find prime factors, to save you time you can use a tool like this to find them for you.

## Isn’t 100 Interesting?

### 100 is a VERY Important Number

100 cents in one dollar, 100 percent makes a whole, water boils at 100°C whereas 100°F is roughly human body temperature.

100 years in a century, the mean adult IQ is 100, and there are 100 senators in the Senate.

100 even crops up in football: the field is 100 yards long!

### Why is 100 Everywhere?

Simply, because it’s a very ‘round’ number: 100 is 10^{2}, and because most of the world’s number systems use base 10 – the value of 10^{2} is very easy to deal with.

However, the word ‘hundred’ wasn’t always used in this way…

Originally, ‘hundred’ meant 120!

It was a unit of measurement used in taxation in medieval England, and various countries in Europe.

But, depending on what was being bought or sold, for some things ’hundred’ meant 120…

…for others, it meant 100 – how confusing!

Base 10 isn’t the only way that numbers can be viewed. Computers deal with binary (base 2) and hexadecimal (base 16), where 100 isn’t nearly as nice: in binary it is 1100100, and in hexadecimal, it is 64.

Can you find different representations of 100 in other bases? Share your answer(s) in the comments below!

### 100 is Extremely Mathematically Interesting Too!

It is the sum of the first 9 prime numbers:

It is also the sum of the first 4 cubes, and the square of the sum from 1 to 4:

= (1 + 2 + 3 + 4)

^{2}

The right-angled triangle below has integer-length sides, with one of length 100:

We call these integer solutions to Pythagoras’ theorem Pythagorean triples.

^{2}+ 612

^{2}= 629

^{2}

Can you find any other Pythagorean triples involving the number 100?

Not only is 100 a square number, but it’s also an abundant number.

A number is called **abundant** if the sum of its factors is greater than twice the number itself.

In the case of 100, the total of it’s divisors is:

It is clear to see that

So 100 *is* an abundant number.

### What’s the Smallest Abundant Number?

Since the only divisors of a prime number are 1 and itself, if *p* is a prime, then the sum of its factors is 1 + *p* < 2*p*, so prime numbers are never abundant! We can check some small numbers, ignoring any primes.

x | 2 x | Sum of Factors | >2x ? |
---|---|---|---|

4 | 8 | 1 + 2 + 4 = 7 | ❌ |

6 | 12 | 1 + 2 + 3 + 6 = 12 | ❌ |

8 | 16 | 1 + 2 + 4 + 8 = 15 | ❌ |

9 | 18 | 1 + 3 + 9 = 13 | ❌ |

10 | 20 | 1 + 2 + 5 + 10 = 18 | ❌ |

12 | 24 | 1 + 2 + 3 + 4 + 6 + 12 = 28 | ✅ |

This means that 12 is the smallest abundant number!

## To Sum Up (Pun Intended!)

To recap, we have found the factors of 100 using divisibility tests. They are:

**1, 2, 4, 5, 10, 20, 25, 50, 100**

And we found the prime factorization of 100 as well. It is:

**100 = 2**

^{2}× 5^{2}Make sure you understand the methods used. Test out what you have learned by factoring some other numbers, using the different methods shown here.

We hope you have enjoyed reading about factorizing 100 more than you expected. If you would like to see a complete table of multiples and factors from 1 to 100, follow the link!

Please comment below with your solutions to any of the challenges, or if you have any questions. We’d also love to hear any other facts you’ve got to share, mathematical or otherwise, about the number 100.

See our other factor lessons:

Factors of 42

Factors of 56