1.6 Real-Life Applications of HCF and LCM

Both come from the same factorisations, so the whole difficulty is choosing which one the question wants.

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.

We need the HCF of $120$ , $84$ and $156$ .
$120 = 2^3 \times 3 \times 5$
$84 = 2^2 \times 3 \times 7$
$156 = 2^2 \times 3 \times 13$
$\text{HCF} = 2^2 \times 3 = 12 \quad \scriptsize\textit{smallest index of each common prime}$
$\therefore$ 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.

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?

$12 = 2^2 \times 3$
$15 = 3 \times 5$
$20 = 2^2 \times 5$
$\text{LCM} = 2^2 \times 3 \times 5 = 60 \quad \scriptsize\textit{highest index of each prime}$
$60$ minutes $= 1$ hour
$\therefore$ All three services next depart together at 8 am

Try it

A practice question on this appears here. Practise primes hcf lcm.

Reverse problem: finding a number from HCF and LCM

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

For any two numbers, $a \times b = (\text{their HCF}) \times (\text{their LCM})$ .
$36 \times b = 12 \times 180$
$36b = 2160$
$b = 60$
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. Write everything out: $36 = 2^2 \times 3^2$ , $\text{HCF} = 2^2 \times 3$ , $\text{LCM} = 2^2 \times 3^2 \times 5$ .
For each prime, the HCF takes the LOWER index and the LCM the HIGHER one, so $b$ is forced prime by prime.
For $2$ : both are $2^2$ , so $b$ has $2^2$ . For $3$ : the lower is $3^1$ and $36$ has $3^2$ , so $b$ has $3^1$ . For $5$ : $36$ has none and the LCM has $5^1$ , so $b$ has $5^1$ .
$\therefore b = 2^2 \times 3 \times 5 = 60$ , the same answer, using only the method this chapter teaches.

Practise this

Questions on Primes, HCF and LCM, marked as you go, with the working shown.

Practise Primes, HCF and LCM