Maths olympiad question · P2–P4★★ Medium

Number puzzles: olympiad question 5

A medium question on number puzzles for P2–P4, with a step-by-step solution, diagrams and the answer. Number puzzles and made-up operations appear in the Hua Xia Cup from the P1 and P2 first rounds to the P6 finals.

The question

May copied out a multiplication, but two digits came out unreadable, shown as □.

Each □ stands for one digit.

3□ × 6 = 2□6

What is the answer to the multiplication?

Show the hints (2)

Hint 1: Start with the ones: which digit times 6 ends in 6?
Hint 2: There is more than one. Multiply out each possibility and see whether the hundreds digit is 2.

Worked solution

Have a go first, then read the solution step by step.

1.Ones: □ × 6 ends in 6, so □ could be 1 or 6.
2.31 × 6 = 186: hundreds digit not 2 — no; 36 × 6 = 216: hundreds digit 2 — yes.
3.Only 36 works, so the answer is 216.
⚠️Common mistake: stopping at the first digit that gives a 6 in the ones, without checking them all.

Answer: 216

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