Lesson 14 of 15

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Eleven ideas, eleven checks. The table is here if one stalls.

Eleven ideas, and the question each one answers about two numbers
Idea	What it says
Primes inside	Every whole number above 1 is one string of primes.
A factor tree	Split it any way. The order changes; the ends do not.
Biggest shared factor	The largest number on both factor lists.
HCF off the primes	Pair the two prime lists, and multiply what pairs up.
Where two counts meet	The first number on both lists of multiples.
LCM off the primes	Pool both prime lists, taking the most of each piece.
Which one is wanted	Goes into both numbers means HCF. Both go into it means LCM.
Check it	Nothing bigger goes into both. Nothing smaller has both going into it.
Co-prime pairs	Two numbers are co-prime when 1 is the only factor they share.
One division table	Stop when the two numbers in a row are co-prime. Multiply the divisors for the HCF. Multiply the divisors and the last row for the LCM.
The product rule	HCF × LCM equals the product of the two numbers.

One student breaks 60 down and gets 2, 2, 3, 5. Another splits it a different way. What can the second get?

The same four numbers, maybe in a different order
Different primes that also multiply to 60
A shorter list, from a better first split

What is the prime factorisation of 29?

None, because 29 is prime
1 × 29
29 on its own, one piece long

Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. What is HCF(12, 18)?

24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. Reading the HCF off those lists, what is it?

12
72, the LCM
6, using one 2 and one 3

Multiples of 4: 4, 8, 12, 16, 20, 24. Multiples of 6: 6, 12, 18, 24. What is LCM(4, 6)?

24, since 4 × 6 = 24
12
There is no lowest one — both lists never stop

12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Reading the LCM off those lists, what is it?

36
216, which is 12 × 18
6, the pieces that pair up

A tailor cuts one strip into 12 cm pieces, nothing left over. A second strip of the same length gives 18 cm pieces. Again, nothing is left over. What is the shortest the strip could be?

The HCF, since the strip is cut into pieces
The LCM
Neither — multiply the two lengths

Someone offers 3 as HCF(12, 18). Which test does it fail?

The first — 3 doesn't go into both
The second — 6 goes into both too, and 6 is bigger
Neither — 3 is the HCF

Which pair is co-prime?

14 and 35
26 and 28
16 and 27

A division table has divisors 2 and 3. Its last row is 5 and 7. What is the LCM?

210, which is 2 × 3 × 5 × 7
35, the last row alone
6, the divisors alone

Two numbers have HCF 4 and LCM 60. One of them is 12. What is the other?

5, since 60 ÷ 12 = 5
15, since 60 ÷ 4 = 15
20, since 4 × 60 ÷ 12 = 20

Your textbook calls this Part II, Chapter 3, Finding Common Ground. You peeled a number and paired primes. You tested answers and built a table.

Next: Another Peek Beyond the Point. Multiplying a decimal, and sharing one out.

Reveal every answer to finish this lesson.

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