007i - Problem Solving with Prime Factorisation

Lesson 7 — Problem Solving and Consolidation: Prime Factorisation

Strand: Number | Descriptor: AC9M7N02 | Duration: 45 minutes

Learning Intentions

• To apply prime factorisation to find highest common factors and lowest common multiples.
• To solve practical problems using the structure of prime factorisations.

Success Criteria

I can:

1. Find the HCF of two numbers by taking the lowest power of each shared prime.
2. Find the LCM of two numbers by taking the highest power of every prime present.
3. Choose whether a problem needs HCF or LCM and justify the choice.
4. Explain my reasoning using index notation.

Warmup

(6 minutes — retrieval, mini whiteboards)

1. Write as a product of powers of primes.
2. Write as a product of powers of primes.
3. What primes do they share? To what power does each appear in both?

Answers:

Shared primes: (to the power in both, taking the lower), (power ), (power ).

Activities

Activity 1 — Explicit Instruction: HCF from Prime Factors (10 min)

Definition. The highest common factor (HCF) of two numbers is the largest number that divides both exactly.

Method: for each prime appearing in both factorisations, take the lower exponent, then multiply.

I do: HCF of and , using the warmup.

Check: ✓ and ✓.

We do: HCF of and .

Note: appears in only one number, so it contributes nothing to the HCF.

You do: HCF of , , .

Activity 2 — Explicit Instruction: LCM from Prime Factors (8 min)

Definition. The lowest common multiple (LCM) of two numbers is the smallest number that both divide into exactly.

Method: take every prime appearing in either factorisation, to its highest exponent, then multiply.

Useful check to teach:

You do: LCM of , , . (Answers: , , .)

Activity 3 — Applied Problems: Which One Do I Need? (12 min)

Pairs. The hard part is deciding between HCF and LCM.

Problem 1 (HCF). A florist has roses and tulips. She wants to make identical bouquets using all the flowers, with no flowers left over. What is the greatest number of bouquets she can make, and what does each contain?

Problem 2 (LCM). Two lighthouses flash at regular intervals. One flashes every seconds, the other every seconds. They flash together at midnight. When do they next flash together?

Problem 3 (LCM, less obvious). Hot dogs come in packs of ; buns come in packs of . What is the smallest number of each pack you must buy so that nothing is left over?

Socratic scaffolding for Problem 1:

Prompt	Purpose
Understand: what is the unknown?	The number of bouquets.
What is the condition?	Every bouquet identical, and no flowers left over.
What does “no flowers left over” tell you about the number of bouquets?	It must divide exactly and divide exactly — a common factor.
Which common factor do we want?	The greatest, since she wants the most bouquets.
Devise a plan	Find .
Carry it out	(from Activity 1).
Answer the actual question	roses and tulips per bouquet.
Looking back	Would bouquets also work? Yes — but it isn’t the greatest. Why does the question ask for greatest?

Answers: 1 — bouquets, each with roses and tulips. 2 — seconds after midnight. 3 — , so packs of hot dogs and packs of buns.

Distinguishing prompt to write on the board:

Splitting things up into equal groups → HCF. Waiting for things to line up / repeat together → LCM.

Checks for Understanding

(5 minutes — exit ticket)

1. Find the HCF of and using prime factorisation. Show your index working.
2. Find the LCM of and .
3. Two bells ring every and minutes. Do you need HCF or LCM to find when they next ring together? Give the answer.
4. Explain why the HCF of two different prime numbers is always .

Answers: 1. , , HCF ; 2. ; 3. LCM — minutes; 4. Distinct primes share no prime factors, so the only common factor is .

Common Misconceptions

Misconception	How to pre-empt it
Swapping HCF and LCM.	Anchor on the words: highest common factor is a factor, so it is smaller than both numbers; the lowest common multiple is a multiple, so it is larger. Sanity-check every answer against this.
Taking the highest exponent for HCF.	Say the reason aloud: a common factor cannot use more copies of a prime than the poorer of the two numbers has.
Forgetting primes that appear in only one number when finding LCM.	Set out both factorisations in aligned columns, writing where a prime is absent.
Assuming HCF is always one of the two numbers.	Counterexample: , which is neither.
Answering the sub-question instead of the actual question (e.g. giving when asked what each bouquet contains).	Build “re-read the question” into the Looking Back step every time.
Thinking coprime numbers have “no HCF”.	Their HCF is — every pair of numbers has an HCF.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two numbers have an HCF of and an LCM of . One of the numbers is . What is the other?

Answer

E2 (Kangaroo style). Three runners complete a lap in , and seconds respectively. They start together. After how many seconds do all three cross the start line together again?

Answer

E3 (Challenge). What is the smallest number that leaves a remainder of when divided by each of , , , and ?

Answer

, so the number is .

E4 (Challenge). A rectangular floor measuring cm by cm is to be tiled with identical square tiles, with no cutting. What is the largest possible tile side length, and how many tiles are needed?

Answer

Tiles are cm. Number needed: tiles.

Homework

1. Use prime factorisation to find the HCF of: (a) and (b) and (c) and (d) and .
2. Find the LCM of: (a) and (b) and (c) and (d) and .
3. Verify that the product of the numbers, for and .
4. Two trains leave a station together. One returns every minutes, the other every minutes. How long until they are both at the station together again?
5. A teacher has pencils and erasers to divide into identical packs with nothing left over. What is the greatest number of packs, and what is in each?
6. Reasoning. Explain why the LCM of two numbers can never be smaller than the larger of the two.
7. Challenge. The HCF of two numbers is and their LCM is . Both numbers are greater than . Find both numbers.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — HCF , LCM ; . Q4 — minutes (3 hours). Q5 — packs, each with pencils and erasers. Q7 — and .