Arithmetic and Number Theory
Core concepts
Arithmetic questions test fluency with fractions, decimals, percentages, ratios, and powers, plus the number-theoretic tools that let you avoid brute computation: divisibility rules, prime factorisation, greatest common divisor (GCD) and least common multiple (LCM), remainders and modular thinking, and the distinction between rational and irrational numbers. Know that every integer factors uniquely into primes, that GCD x LCM = product of the two numbers, and that a fraction is in lowest terms exactly when numerator and denominator share no prime factor.
Traps to watch
- Percentage change is not symmetric: a 20% rise followed by a 20% fall leaves 96% of the original, not 100%.
- Percentage points versus percent: going from 25% to 30% is a 5 percentage-point gain but a 20% relative increase.
- Dividing by a fraction smaller than 1 increases the value; many candidates multiply instead.
- Remainder problems: negative or large powers need modular reduction, not direct computation.
- Units and rounding: check whether the question asks for exact or approximate answers.
Rule of thumb
For LCM/GCD word problems (bells ringing together, tiles fitting a rectangle), ask whether the question wants the next simultaneous event (LCM) or the largest equal grouping (GCD).
Worked example
Find the smallest positive integer n such that n leaves remainder 3 on division by 8 and remainder 5 on division by 9. Candidates are 3, 11, 19, 27, 35, 43; checking mod 9: 3, 2, 1, 0, 8, 7 - none yet. Continue: 51 gives 6, 59 gives 5. So n = 59.