2025 HSC Mathematics Extension 1
Question 14(a):
Pigeonhole principle
Prove that the product of any seven distinct factors of 60 is divisible by 60.
How to recognise this question
Question type: pigeonhole principle
- The command and notation point to pigeonhole principle.
- The requested response is a show that.
How to handle it: identify the objects and boxes, then show why avoiding a repeated box is impossible.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Pair the twelve factors into six buckets: .
- Choosing seven distinct factors forces two to come from one bucket.
- Those two multiply to , so the product of all seven chosen factors is a multiple of .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Prove that among any six integers, two have a difference divisible by 5.
Question 2
A drawer contains red, blue and green socks. How many socks guarantee four of one colour?
Question 3
Show that choosing six numbers from guarantees two consecutive numbers.