Cutting ribbons
Two ribbons, 12 cm and 18 cm long, are cut into pieces that are all the same length, with nothing left over. What is the longest the pieces can be?
Each piece has to fit exactly into 12 and exactly into 18, so its length divides both numbers. The longest length that divides both is the Highest Common Factor: the HCF of 12 and 18 is 6, so the longest piece is 6 cm.
Pieces of 6 cm: the 12 cm ribbon makes 2 and the 18 cm ribbon makes 3, with nothing left over.
Buses leaving together
Two buses leave a station together. One leaves every 4 minutes and the other every 6 minutes. How long is it until they next leave together?
The first bus leaves at 4, 8, 12, 16 minutes, and the second at 6, 12, 18 minutes. The time they share must be a multiple of 4 and a multiple of 6, so both numbers divide it. The first time both divide is the Lowest Common Multiple: the LCM of 4 and 6 is 12, so they leave together again after 12 minutes.
The first bus leaves at 4, 8 and 12 minutes.
The second bus leaves at 6 and 12 minutes. 12 is the first time on both lines.
Which way does the dividing go?
Both questions give two numbers and ask for a third. What decides between the HCF and the LCM is which way the dividing goes.
In the ribbon question the answer divides the given numbers: 6 divides 12 and 6 divides 18. An answer that divides both numbers is the HCF.
In the bus question the given numbers divide the answer: 4 divides 12 and 6 divides 12. An answer that both numbers divide is the LCM.
So ask: must the answer fit into both numbers, or must both numbers fit into it? This works for any story, including ones that are not about cutting or meeting.
The size of the answer is a check
The HCF divides both numbers, so it is never larger than the smaller of them: 6 is less than 12 and less than 18. The LCM is divided by both numbers, so it is never smaller than the larger of them: 12 is more than 4 and more than 6.
This catches the wrong choice straight away. A piece 36 cm long, the LCM of 12 and 18, could not be cut from a 12 cm ribbon at all. A meeting after 2 minutes, the HCF of 4 and 6, would come before either bus had left a second time.
The HCF of 12 and 18 sits below both numbers.
The LCM of 4 and 6 sits above both numbers.
More stories
Two boxes, of 20 and of 30 sweets, are shared into bags of equal size with none left over. What is the biggest bag? The bag size must fit into 20 and into 30, so it is the HCF: 10 sweets.
Two lights flash together, then one flashes every 3 seconds and the other every 5 seconds. When do they flash together again? 3 and 5 must both fit into the time, so it is the LCM: 15 seconds.
Buns come in packs of 6 and sausages in packs of 8, and the same number of each is needed. The number must be reached by counting in 6s and by counting in 8s, so it is the LCM: 24.
Read the question, not the key words
Words such as "every" or "equal" appear in both kinds of question, so they cannot decide. Decide by the dividing: does the answer divide the numbers, or do the numbers divide the answer?
Multiplying the two numbers answers neither question. 12 × 18 = 216 is far too long to cut from either ribbon, and 4 × 6 = 24 minutes is a time the buses do share, but not the first one.