A hard question on cycles for P4–P5, with a step-by-step solution, diagrams and the answer. Cycle questions appear in the Hua Xia Cup from the P1 first round to the P5 papers: which comes in a given place, which day of the week, which last digit.
The question
Multiply 1843 copies of 9 together to get a huge number A, and 531 copies of 4 to get another huge number B.
A = 9 × 9 × 9 × … × 9B = 4 × 4 × 4 × … × 4
What is the last digit of A × B?
Show the hints (3)
💡Hint 1: Look only at last digits: what pattern do 9, 9 × 9, 9 × 9 × 9, … follow?
💡Hint 2: Write out the last digits of 9, 9 × 9, … until they repeat: how long is the cycle? Divide 1843 by that length and use the remainder to find A's last digit.
💡Hint 3: Find B's last digit the same way (B has 531 copies of 4, not 1843). Then just multiply the two last digits and take the last digit of that.
Worked solution
Have a go first, then read the solution step by step.
1.Products of 9 end in 9, 1, repeating every 2.
2.1843 ÷ 2 = 921 remainder 1, so it is the 1st: A ends in 9.
3.Products of 4 end in 4, 6, repeating every 2.
4.531 ÷ 2 = 265 remainder 1, so it is the 1st: B ends in 4.
5.Only the last digits matter: 9 × 4 = 36, which ends in 6.
⚠️Common mistake: assuming a product of 9s always ends in 9.
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