Chapter 3: Review

The chapter problem, solved

Two lighthouses flash every 12 seconds and every 18 seconds. They flash together. How long until they next flash together?

The question asks when two repeating events line up again. That happens at a common multiple of 12 and 18, and "next" makes it the smallest one.

12 = 2^2\times3 \qquad 18 = 2\times3^2

A common multiple has to be big enough to build both numbers, so it takes the larger power of each prime that appears:

\text{LCM} = 2^2\times3^2 = 4\times9 = 36

They next flash together after 36 seconds.

Two checks, and both are free. The answer is bigger than 18, which a lowest common multiple must be. And $36\div12=3$ and $36\div18=2$ are both whole numbers, so 36 really is a multiple of both.

Now the other half of the chapter problem. The shop's 48 blue and 60 black pens go into boxes of one colour, every box the same size, nothing left over. That asks for a number that divides both, so it is a highest common factor.

48 = 2^4\times3 \qquad 60 = 2^2\times3\times5

\text{HCF} = 2^2\times3 = 12

A box holds 12 pens. The two problems gave the same-looking numbers and wanted opposite things — which is the point of the chapter.

Mixed practice — choose the method first

Work through these without notes. For each one, decide which idea applies and say why in a few words. Recognising which of the two ideas a question wants is the skill; the arithmetic follows from it.

Write 252 as a product of its prime factors.
Find the highest common factor of 252 and 90.
Find the lowest common multiple of 252 and 90. Check your two answers against the sizes of the numbers before you write them down.
Is 253 prime? Give a reason.
Two alarms sound every 45 seconds and every 60 seconds. They sound together. When do they next sound together, and which idea did you use?
Squares of card are 84 mm and 126 mm. They are cut into equal squares with nothing wasted, as large as possible. What is the side length of each small square?
How many factors does 252 have? Use its factorisation, not a list.
A number has exactly three factors. What can you say about it?
Work out $2^4\times3^2$ and say whether the result is the highest common factor or the lowest common multiple of 48 and 36.
Explain why the highest common factor of two numbers can never be larger than either number, and why the lowest common multiple can never be smaller.

What you can now do

list every factor of a number, and every multiple up to a given point;
test divisibility by 2, 3, 4, 5, 6, 9 and 10, and say which rule you used;
write any whole number as a product of prime factors in index form, and check it by multiplying back out;
count the factors of a number from its factorisation;
find the highest common factor and the lowest common multiple of two or more numbers, and tell from the wording which one a question wants.

Thinking back

Look at what you wrote at the start of the chapter, where you described something in your own week that has to line up — a bus and a train, two timers, two rotas.

Now that you can do the calculation, work it out. Find the two periods, name them in seconds or minutes, and find when they next line up together. If the real answer surprises you, that is worth writing down: it usually means one of the two periods is not quite what the timetable claims.