Number System

Read the formula sheet first, then work through the 15 questions below. Each answer comes with a step-by-step solution — read it even when you get the question right, the shortcut is often faster than your method.

Formula sheet

• Divisibility: by 3 → digit sum divisible by 3; by 9 → digit sum by 9; by 4 → last two digits; by 8 → last three; by 11 → (sum of odd-place digits) − (sum of even-place digits) is 0 or a multiple of 11.
• Division rule: Dividend = Divisor × Quotient + Remainder. If N leaves remainder r on dividing by D, then N mod d = r mod d for any factor d of D.
• Unit-digit cycles: powers of 2 → 2,4,8,6; 3 → 3,9,7,1; 7 → 7,9,3,1; 8 → 8,4,2,6 (cycle length 4). Digits 0,1,5,6 never change. Use (power mod 4).
• Sum of first n naturals = n(n+1)/2; first n odds = n²; first n evens = n(n+1); Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².
• Trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
• (a+b)(a−b) = a²−b²; a²+b² = (a+b)²−2ab; a³+b³ = (a+b)(a²−ab+b²).

Practice

Pick an answer to see whether it is right and the worked solution. Your score is saved when you finish the set.