An expert question on divisibility and factors for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P4 and P5 qualifying rounds and finals; harder ones use how many factors a number has, and square numbers.
The question
Ming draws 3 number cards:
8, 29 and 56.
He multiplies the numbers together and gets a very big number.
His teacher says: multiply it by one more whole number and it becomes a perfect square (a whole number times itself, like 36 = 6 × 6).
What is the smallest whole number he can multiply by?
Show the hints (3)
💡Hint 1: When a perfect square is written as a product of primes, what do you notice about how often each prime appears?
💡Hint 2: No need to multiply it out: write each card's number as a product of primes.
💡Hint 3: Count how many times each prime appears altogether; each one appearing an odd number of times needs one more.
Worked solution
Have a go first, then read the solution step by step.
1.In a perfect square every prime appears an even number of times: 36 = 6 × 6 = 2 × 2 × 3 × 3, the 2s and 3s in pairs.
4.The odd ones are 7, 29; one more of each makes every count even: 7 × 29 = 203.
⚠️Common mistake: multiplying by each card that isn't a square on its own, without pooling the primes first, so primes that already pair up get multiplied in again.
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download