HCF and LCM for Class 5 — Listing Method

The Story

Ms. Rao put two problems on the board, both using 12 and 18.

First: a strip of card 12 cm and another 18 cm, cut into equal pieces with none wasted. What is the longest piece?

Second: two bells, one every 12 minutes and one every 18. When do they ring together?

Kabir looked at both and said, "Same numbers, so same method."

He listed the factors of 12: 1, 2, 3, 4, 6, 12. Then of 18: 1, 2, 3, 6, 9, 18. Shared: 1, 2, 3, 6. Biggest shared: 6.

He wrote 6 for both problems.

"Test the second one," said Ms. Rao. "Bells every 12 and 18 minutes. Do they both ring at 6 minutes?"

Kabir checked. The first bell rings at 12, 24, 36. The second at 18, 36. Neither rings at 6.

"Six is smaller than both," said Anaya. "A bell can't ring before its first ring."

"So which answer must be bigger than both numbers, and which must be smaller?"

Kabir saw it. "The tile has to fit INSIDE, so it's smaller. The bells have to wait for BOTH, so it's later than both."

He listed the multiples instead. 12: 12, 24, 36, 48. 18: 18, 36, 54. First shared: 36.

"Thirty-six minutes for the bells. Six centimetres for the card."

"And here is the check that costs nothing," said Ms. Rao. "Your HCF can never be bigger than the smaller number. Your LCM can never be smaller than the larger one. If either of those goes wrong, you have picked the wrong one before you even look at the arithmetic."

Anaya wrote it in her book as two arrows: HCF points down and inward, LCM points up and outward.