Applications of the Fundamental Theorem of Arithmetic - 4

Highest Common Factors and Lowest Common Multiples

Firstly, note that the highest common factor (HCF) of two or more numbers is also commonly called the greatest common divisor (GCD). Either way, it represents the highest/greatest number by which two (or more) other numbers can be divided.

It should be fairly obvious that the HCF of 4 and 6 is 2.
2 is the largest number that can divide both 4 and 6.
4 ÷ 2 = 2
6 ÷ 2 = 3

By observation, we can look at each equation in reverse to see that:
2×2 = 4
3×2 = 6

Have you spotted it?

2×2, or rather, 2², is the prime factorisation of 4;

3×2, or rather, 2×3, is the prime factorisation of 6.

2 is the highest number in both prime factorisations, which is why it is the HCF of 4 and 6.

The lowest common multiple (LCM, sometimes calles the least common multiple) of two or more numbers is the lowest/smallest number which is in the times table of the numbers.

Again, using 4 and 6, it should be fairly obvious that the LCM is 12.

12 is the smallest number which is in the 4 times table and the 6 times table.

One way this can be seen is to write out the times tables:

4 times table: 4     8     12

6 times table: 6     12

This can also be calculated using the Fundamental Theorem of Arithmetic.

We will take two numbers: 60 and 144
Find the prime factorisation of each one (in expanded form rather than index form):
60 = 2×2×3×5
144 = 2×2×2×2×3×3

Identify the numbers that appear in both lists: each list has two 2s and one 3.
60 = 2×2×3×5
144 = 2×2×2×2×3×3

Since each list contains 2×2×3, this means that:
HCF = 2×2×3 = 12

Since we have identified what is common to both lists of prime factors, the LCM must also include the numbers which are not common to both lists. In other words, we need 2×2×3 (which is common to both), but also 5×2×2×3 (which is not common to both, appearing in only one of the lists).
LCM = 2×2×3×5×2×2×3 = 720

Another way of looking at this is to just take one number and multiply it by the non-highlighted numbers in the list of the other number's prime factorisation:
LCM = 60×2×2×3 = 720
or
LCM = 144×5 = 720

Can you find the HCF and LCM of 20 and 30?
What about 56 and 84?
210 and 175?

It is also possible to use a very similar method to find the HCF and LCM of three (or more) numbers.
Can you find the HCF and LCM of 24, 84 and 150?

Applications of the Fundamental Theorem of Arithmetic - 3

The number of factors

Not only does the Fundamental Theorem of Arithmetic help us to find all the factors of a number, it is a quick way to know how many factors a number has in total.

Take the number 24:

1     24

2     12

3     8

4     6

So 24 has 8 factors.

Written as prime factors:

24 = 2³×3

If we take the powers: 3 and 1

Increase them each by one: 4 and 2

Then multiply them together: 4×2 = 8

So 24 has 8 factors.

Try another number: 180

180 = 2²×3²×5

The powers are 2, 2 and 1.

Increase them each by 1 to get 3, 3 and 2.

Multiply them together: 3×3×2 = 18

So 180 should have 18 factors.

This can be checked by working them out:

180

1     180

2     90

3     30

4     45

5     36

6     30

9     20

10     18

12     15

Can you find out how many factors 48 has?

What about 216?

Or 1260?

Can you list all the factors?

Applications of the Fundamental Theorem of Arithmetic - 2

Finding all the factors of a number.

Hopefully it is fairly obvious that the factors of 12 are 1, 2, 3, 4, 6 and 12.
Personlly, I like to have them written down in pairs, as it helps to know that you've found them all:

12
1     12
2     6
3     4

Using the Fundamental Theorem of Arithmetic:

12 = 2²×3

or, in expanded form:

12 = 2×2×3

How does this help? Well, 1 is a factor of every number, so we include it anyway. Every other factor of 12 can be made with every different combination of its prime factors:

1 - factor of every number

2 - 2

3 - 3

4 - 2×2

6 - 2×3

12 - 2×2×3

To find all the factors of 72:

72 = 2³×3²

or, in expanded form:

72 = 2×2×2×3×3

So, the possible factors are:

1 - factor of every number

2 - 2

3 - 3

4 - 2×2

6 - 2×3

8 - 2×2×2

9 - 3×3

12 - 2×2×3

18 - 2×3×3

24 - 2×2×2×3

36 - 2×2×3×3

72 - 2×2×2×3×3

The interesting thing about this method is that all of the factors are written as products of prime numbers too.

Can you use this method to find all the factors of 54?

What about 105?

Or 120?

Applications of the Fundamental Theorem of Arithmetic - 1

Finding square numbers.

Let's say we have a number, 234, and we want to know what we need to multiply it by to make a square number.

As a product of prime factors:

234 = 2×3²×13

To make a square number, the powers need to be even.
The power of 3 is already even.
To make the power of 2 even, we need to multiply by another 2.
Similarly for 13.

2×13 = 26

So if we multiply 234 by 26, we will get a square number.

234×26 = 6084

and 6084 = 78²

It is also important to note that:

78 = 2×3×13

Have you noticed the pattern? This is just the prime factors of 6084, but halving each of the powers.

Can you work out what you need to multiply 112 by to make a square number? 

What about 7425?

Can you find a similar pattern for making cubic numbers?

The Fundamental Theorem of Arithmetic

This is possibly the most important aspect of numbers:

Every positive integer (whole number), except for the number 1, is either a prime number or can be written as a product of prime factors.

Here are the numbers up to 10:

2 – prime

3 – prime

4 – 2²

5 – prime

6 – 2×3

7 – prime

8 – 2³

9 – 3²

10 – 2×5

See if you can continue this list up to 20 or 30.

Why is this so important? Well, it can help us to find square numbers:

36 = 2²×3²                   a square number

81 = 3⁴                         a square number

144 = 2⁴×3²                 a square number

400 = 2⁴×5²                 a square number

3969 = 3⁴×7²               a square number

4356 = 2²×3²×11²        a square number

Have you spotted the pattern? All the powers are multiples of two.

Can you find a similar pattern for cubic numbers?

What about quartic numbers?

Quintic numbers?

Note: writing numbers as shown above is called 'writing as a product of prime factors in index form'.

Further reading: https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

Purpose

This blog is for people who wish to take a plunge into the fascinating world of mathematics, exploring the logic and beauty of numbers, pushing their skills further.

Initially, it will be aimed for secondary school pupils pushing the boundaries of GCSE, but may well extend to much more!