Arithmetic: Number Systems, HCF/LCM, Fractions and Decimals
What you must know
Start with divisibility rules (2,3,4,5,6,8,9,11), prime factorisation, and the classification of numbers (natural, integer, rational, irrational). Every HCF/LCM question becomes easy once you factorise: HCF takes the lowest power of common primes, LCM takes the highest power of all primes. Remember the anchor identity HCF × LCM = product of the two numbers (valid for exactly two numbers).
For fractions, convert to a common denominator before comparing, and remember that dividing by a fraction means multiplying by its reciprocal. Decimal-to-fraction conversion follows place value: 0.375 = 375/1000 = 3/8. Recurring decimals use the 9s trick: 0.̅4̅ = 4/9, 0.1̅2̅ = 12/99 = 4/33.
Traps
- LCM of fractions = LCM of numerators / HCF of denominators; HCF of fractions is the reverse. Candidates routinely invert this.
- "Greatest number that divides a, b, c leaving the same remainder" = HCF of (a−b), (b−c), (a−c).
- "Least number which when divided by a, b, c leaves remainder r" = LCM(a,b,c) + r; but if it must leave different remainders, adjust by the difference.
- Do not cancel across an addition sign: (a+b)/c is not a/c + b/c unless you actually split it correctly.
Rule of thumb and example
When a question mentions bells ringing together, runners meeting at a start point, or tiles fitting exactly, think LCM. When it mentions largest groups, largest measuring rod, or equal distribution, think HCF.
Worked example: Find the least number of soldiers that can be arranged in rows of 12, 15 or 18 with none left over. LCM(12,15,18): 12=2²·3, 15=3·5, 18=2·3², so LCM = 2²·3²·5 = 180. Answer: 180.