### Overview

This section covers Lowest Common Multiples, Highest Common Factors, Squares, Cubes, Roots and Prime Factors. So there are a number of terms to learn and some techniques. Some of these techniques need to be practiced a few times such as using prime factors to find the highest common factor and the lowest common multiple.

### Factors, Powers and Roots Questions

1. Write 225 as a product of prime factors.

2. Write 165 as a product of prime factors.

3. Find the highest common factor of 225 and 165.

4. Write 28 as a product of prime factors.

5. Write 36 as a product of prime factors.

6. Find the lowest common multiple of 28 and 36.

7. Find the highest common factor of 1,368 and 1,512.

8. Manchester United’s groundsman has to relay the practice pitch. The pitch measures 128 metres by 80 metres. He can only buy turf in square units. What is the largest size of square turfs he can buy to cover the pitch exactly?

9. Find the highest common factor of 128, 240 and 360.

10. Buses 1,2 & 3 share the same bus stand at the central bus terminal. All three buses set out at the same time, 7.30 a.m. Bus 1 returns to the terminal every 30 minutes, Bus 2 returns every 40 minutes and Bus 3 returns every 60 minutes. At what time will all 3 buses return to the terminal at the same time (assuming all buses keep to their timetable).

### Factors, Powers and Roots Approach

**1,2,4 & 5.** To find a prime factor keep dividing the number down until you reach prime numbers. Use a factor tree as in the following example to find the prime factors of 40:

So 2 x 2 x 2 x 5 = 40. This is usually shortened using index notation = 2³ x 5.

**3, 7, & 9.** There is a set method to find the highest common factor:

a) Write the prime factors of each number (shortened using index notation where possible)

b) Highlight the prime factors that are common. Highlight the lowest power. For example if you have 2² and 2³ then highlight 2².

c) Mulitiply these common prime factors to give the highest common factor.

You need to learn this method and practice it so you can just do it without having to think about the logic behind it.

I must admit I was a bit stumped about WHY this worked. I think mathematicians would see it as elegant or beautiful! It’s best explained with an example: What is the highest common factor of 36 and 48? If you use the factor tree you get:

2^{4} x 3 = 48

2² x 3² = 36

Using the method above the common prime factors are 3 and 2² (3 is a subset of 3² and 2² is a subset of 2^{4}).

3 x 2² = 12. So if you replace 3 x 2² in the above sums with 12 you get:

2² x 12 = 48

3 x 12 =36

I hope this helps more than it confuses. The key point for Maths GCSE is to learn the method and practice it. I also found this video which helped me: Greatest Common Factor. Please note in the USA highest common factor seems to be referred to as greatest common factor.

**6.** Just as there is a method to find the highest common factor using prime factors, there is also a method to find the lowest common multiple using prime factors.

a) Write the prime factors of each number (shortened using index notation where possible)

b) Highlight the highest power of each prime factor.

c) Mulitiply these factors to give the lowest common multiple.

I think I’ve just about my head round why this works! However I find it very difficult to put into words– probably means my understanding is a bit shaky. If anybody can explain it clearly please let me know!

Anyway the key thing is to learn the method and practice it!

**8.** This is an example of a functional question involving highest common factor. You need to find the HCF to find the largest squares that can exactly fit the pitch dimensions.

**10.** This is a functional question involving lowest common multipliers.

### Factors, Powers and Roots Answers

1.

2.

3.

4.

Prime factors of 28 = 2² x 7

5.

Prime factors of 36 = 2² x 3²

6.

7.

8.

So the largest size turfs he can buy to cover the pitch exactly are 16m²

9.

10.

The first time that the buses should arrive back together at the bus terminal is 7.30 am. plus 120 minutes = 9.30 a.m.