1.6Real-Life Applications of HCF and LCM

Word problems rarely say HCF or LCM. They describe a situation, and you decide.

HCF or LCM? The decision in one table

Both come from the same prime factorisation, so the whole difficulty is choosing which.

	HCF	LCM
The situation	dividing or cutting into equal parts	repeating events coinciding
Words to look for	largest, greatest, biggest possible, equally distributed	next time together, at the same time again, smallest number divisible by
Your answer is	smaller than or equal to the given numbers	larger than or equal to them
If it looks wrong	too large for a largest-piece question means you used LCM	too small for a next-coincide question means you used HCF

Sorting stickers into equal trays

Mei Lin has 120 cat stickers, 84 dog stickers and 156 rabbit stickers. She wants to sort them into identical trays, each holding the same number of cat, dog and rabbit stickers, with none left over. Find the greatest number of trays she can fill, and state what each tray holds.

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We need the HCF of

120

84

and

156

\displaystyle 120

\displaystyle = 2^3 \times 3 \times 5

\displaystyle 84

\displaystyle = 2^2 \times 3 \times 7

\displaystyle 156

\displaystyle = 2^2 \times 3 \times 13

\displaystyle \text{HCF}

\displaystyle = 2^2 \times 3

\displaystyle = 12 \quad \scriptsize\textit{smallest index of each common prime}

\therefore

The greatest number of trays is

12

Each tray:

\frac{120}{12} = 10

cat,

\frac{84}{12} = 7

dog,

\frac{156}{12} = 13

rabbit stickers.

Three services from one interchange.

60

is the first minute all three share, which is the LCM.

Bus timetable problem

Three bus services depart from the same interchange. The first departs every 12 minutes, the second every 15 minutes and the third every 20 minutes. All three services depart together at 7 am. When will they next depart together?

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\displaystyle 12

\displaystyle = 2^2 \times 3

\displaystyle 15

\displaystyle = 3 \times 5

\displaystyle 20

\displaystyle = 2^2 \times 5

\displaystyle \text{LCM}

\displaystyle = 2^2 \times 3 \times 5

\displaystyle = 60 \quad \scriptsize\textit{highest index of each prime}

60

minutes

= 1

hour

\therefore

All three services next depart together at 8 am

Now you try

Bus A departs every $9$ minutes, Bus B departs every $15$ minutes. If both depart at 9 am, after how many minutes will they next depart at the same time?

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Find the LCM of

9

and

15

\displaystyle 9

\displaystyle = 3^{2} \quad \scriptsize\textit{prime factorisation}

\displaystyle 15

\displaystyle = 3 \times 5 \quad \scriptsize\textit{prime factorisation}

\displaystyle \text{LCM}

\displaystyle = 3^{2} \times 5 \quad \scriptsize\textit{highest powers}

\displaystyle = 45

They coincide at every common multiple, so the first one after the start is the LCM.

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Reverse problem: finding a number from HCF and LCM

The HCF of $\displaystyle a$ and $\displaystyle b$ is $\displaystyle 12$ , and their LCM is $\displaystyle 180$ . Given that $\displaystyle a = 36$ , find $\displaystyle b$ .

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For any two numbers,

a \times b = (\text{their HCF}) \times (\text{their LCM})

\displaystyle 36 \times b

\displaystyle = 12 \times 180

\displaystyle 36b

\displaystyle = 2160

\displaystyle b

\displaystyle = 60

Check:

36 = 2^2 \times 3^2

60 = 2^2 \times 3 \times 5

. HCF

= 2^2 \times 3 = 12

✓, LCM

= 2^2 \times 3^2 \times 5 = 180

✓

Without the formula, index notation gets there too. Line the three up prime by prime:

prime	$36$	HCF $=12$	LCM $=180$	so $b$
$2$	$2^2$	$2^2$	$2^2$	$2^2$
$3$	$3^2$	$3^1$	$3^2$	$3^1$
$5$	none	none	$5^1$	$5^1$

Read each row across: the HCF takes the lower index of the two numbers and the LCM takes the higher one.

36

is known, so every index of

b

is forced.

\therefore b = 2^2 \times 3 \times 5

= 60

, the same answer.