Challenge Questions – Factors, Multiples and Primes
These problems are designed to stretch your thinking. Each one requires more than one concept from this learning path. Attempt them before reading the solutions.
Problems like these — where you have to combine two or three ideas rather than apply a single rule — are exactly the style used in maths olympiad and competition settings for younger students. There is rarely a single “obvious” first step; part of the challenge is deciding which piece of information to use first. If you get stuck, try listing everything you know for certain before attempting to combine it.
Challenge 1 – Finding a Mystery Number
I am a two-digit number. I am a multiple of 8. My digits add up to 9. I am not a perfect square. What am I?
Multiples of 8 up to 99: 8,16,24,32,40,48,56,64,72,80,88,96. Digit sums = 9: 72 (7+2=9) and 90 (9+0=9, not mult of 8). Is 72 a perfect square? √72 ≈ 8.5 — No. Answer: 72.
Challenge 2 – The Scheduling Problem
Three buses leave a terminal every 12, 15, and 20 minutes. They all depart together at 6:00 am. When is the next time all three leave together?
LCM(12,15,20): 12=2²×3; 15=3×5; 20=2²×5. LCM=2²×3×5=60 minutes. Next joint departure: 7:00 am.
Challenge 3 – The Tile Problem
A rectangular courtyard 504 cm × 336 cm is to be paved with identical square tiles with no cutting. (a) What is the side length of the largest possible tile? (b) How many tiles are needed?
(a) GCF(504,336): 504=2³×3²×7; 336=2⁴×3×7. GCF=2³×3×7=168 cm.
(b) Tiles = (504÷168) × (336÷168) = 3 × 2 = 6 tiles.
Challenge 4 – Counting Factors
A number N = 2² × 3 × 5². How many factors does N have? List them all.
Number of factors = (2+1)(1+1)(2+1) = 18. They are: 1,2,3,4,5,6,10,12,15,20,25,30,50,60,75,100,150,300.
More Challenges
- Find two numbers whose GCF is 12 and LCM is 180.
- Is 2³ × 3² × 7 prime? How many factors does it have?
- The GCF of two consecutive even numbers is always what? Prove it.
- How many prime numbers less than 100 end in 3?
- Write 10,000 as a product of prime factors and count its factors.
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