Playbook
Number System
223 q · 16% HARD · 13 a paper recently, up from 9.8. The HARD is spread across remainders, factorisation, HCF laws and division rather than pooled, so there is no page to leave out; primes, HCF/LCM applications and irrationals have never set a HARD question.
- Questions in the bank
- 223
- q/paper (21 papers)
- 10.62
- Tagged HARD
- 16%
- Subtopics
- 12
Strand: Cornerstone
When you’ll see it
Divisibility, remainders, factors, HCF and LCM, unit digits, or a digit puzzle.
How this chapter is tested
223 questions, level with Trigonometry on recent papers at 13 a paper (up from 9.8 in 2016-2020). It is twelve subtopics of whole-number reasoning, and almost every question has one short route if you know the right rule.
Its HARD (16%) is spread rather than pooled: remainders by congruence, divisibility by factorisation, HCF laws and division each carry some. Primes, HCF and LCM applications and irrational numbers have never set a HARD question. There is no page to leave for last — own the chapter.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Divisibility rules
By 3, 4, 8, 9, 11 and composites through their co-prime factors.
Factors and trailing zeros
Divisor count from prime powers; trailing zeros of n! from powers of 5.
HCF and LCM
HCF × LCM = product of two numbers; remainder recipes for 'leaves remainder r in each case'.
Remainders
Reduce the base mod n, then use cyclicity or (a ± 1)ⁿ.
Unit digits
Powers cycle in blocks of four; take the exponent mod 4.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.
HCF × LCM for three numbers
The product rule holds for two numbers only.
Divisibility by a product needs co-prime factors
Divisible by 3 and by 9 is only divisible by 9, not 27; divisible by 2 and by 4 need not be divisible by 8. Test 12 as 3 × 4, not 2 × 6.
1 is neither prime nor composite
Counting it in either set shifts the answer by one.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each subtopic. Read the notes once, then drill subtopic by subtopic below.
Number System notesDrill every Number System question
223 questions from the bank, across 12 subtopics.
Drill one subtopic at a time
The 12 subtopics, in teaching order.
- Division, Parity and Consecutive IntegersDrill Division, Parity and Consecutive Integers
- Place Value and Digit ProblemsDrill Place Value and Digit Problems
- Divisibility Rules and Missing DigitsDrill Divisibility Rules and Missing Digits
- Unit Digit and CyclicityDrill Unit Digit and Cyclicity
- Prime Numbers and PrimalityDrill Prime Numbers and Primality
- Factors, Divisor Counting and Trailing ZerosDrill Factors, Divisor Counting and Trailing Zeros
- HCF and LCM Laws and FractionsDrill HCF and LCM Laws and Fractions
- HCF and LCM Applications and Remainder RecipesDrill HCF and LCM Applications and Remainder Recipes
- Remainders by Congruence and CyclicityDrill Remainders by Congruence and Cyclicity
- Divisibility by FactorisationDrill Divisibility by Factorisation
- Perfect Squares, Cubes and Difference of SquaresDrill Perfect Squares, Cubes and Difference of Squares
- Rational and Irrational NumbersDrill Rational and Irrational Numbers
Related playbooks
Often paired with this one — the technique or the trap overlaps. Drill these next.