Non-routine problem solving (Grade 8)
What Grade 8 tests
Grade 8 problems reward structure: cycles in powers, symmetry in counting, similarity ratios, and algebraic factoring. The calculation is rarely the hard part; recognizing the pattern is.
Core rules
- Modular cycles: compute successive powers modulo m until the remainders repeat. For modulus 5, powers of 7 behave like powers of 2: 2, 4, 3, 1, then repeat every 4.
- Probability without replacement: multiply changing fractions. Drawing two reds from 3 red and 2 blue gives (3/5) × (2/4) = 3/10.
- Similar figures: corresponding lengths scale by k, areas by k², volumes by k³.
- Quadratics: factor, complete the square, or use the discriminant to count real solutions.
- Invariants: parity, digit sums, and remainders often rule out impossible configurations faster than exhaustive listing.
Worked example
Find the remainder when 7^2024 is divided by 5. Since 7 ≡ 2 (mod 5), we need 2^2024 mod 5. The powers of 2 modulo 5 cycle: 2^1=2, 2^2=4, 2^3=3, 2^4=1, then repeat with period 4. Divide the exponent by 4: 2024 = 4 × 506, so the exponent is a multiple of 4. The cycle position is 4, where the remainder is 1. Answer: 1.
Traps
- Reducing the base but not the exponent.
- Using the wrong cycle length after the remainders start repeating.
- Treating draws without replacement as independent.
- Scaling area by k instead of k² in a similarity problem.