A medium question on remainders for P4–P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 papers; harder ones need several remainder conditions to hold at once.
The question
Ming plays a phone game.
He starts with 1 coin, and every level he clears multiplies his coins by 3.
He clears 26 levels.
What is the last digit of his number of coins now?
Show the hints (2)
💡Hint 1: Try the first few levels: what is the last digit after each one? Is there a pattern?
💡Hint 2: Every how many levels does the last digit repeat? Divide 26 by that number and look at the remainder.
Worked solution
Have a go first, then read the solution step by step.
1.Each last digit depends only on the previous last digit, so track last digits only.
2.From level 1 the last digits go 3, 9, 7, 1, 3, … repeating every 4 levels.
3.26 ÷ 4 = 6 remainder 2, the same as level 2: 9.
⚠️Common mistake: using the quotient 6 of 26 ÷ 4 instead of the remainder.
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