# Factors, Powers and Roots

### 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:

24 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 24).

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.

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