1.2.1 Number system and simplification notes

1. 1.2.1.1Number types and divisibility rules

   Know which family a number belongs to and test divisibility from its digits, so that "find the missing digit", "nearest multiple" and "how many numbers between" questions take seconds.

   Advanced · 8 min · Read here
2. 1.2.1.2Remainders, unit digits and factors

   Unit digits repeat in short cycles, remainders can be worked on small pieces, and the number of factors comes straight from the prime factorisation. (Factorials, factor counting and remainder theorems each have a fuller note in this unit: MAB-NS-09, 10 and 11.)

   Advanced · 6 min · Read here
3. 1.2.1.3HCF and LCM: methods and word problems

   HCF is the largest number that divides all the given numbers, LCM is the smallest number they all divide, and most word problems only ask which of the two is needed and how the remainder is adjusted.

   Advanced · 7 min · Read here
4. 1.2.1.4Simplification and approximation: BODMAS, surds, indices and roots

   Simplification questions test order of operations, a handful of identities and the habit of spotting a pattern; most collapse to a small whole number once the pattern is seen. (Decimals, roots, and surds and indices have fuller notes: MAB-NS-06, 07 and 08.)

   Advanced · 6 min · Read here
5. 1.2.1.5Number puzzles: digits, sums and differences

   Write a number in place-value form, turn the words into one or two equations, and most digit puzzles solve in two lines. (Counting digits and page numbers is in MAB-NS-14; general word problems are in MAB-NS-12.)

   Advanced · 6 min · Read here
6. 1.2.1.6Decimal fractions and recurring decimals: conversion and comparison

   Every fraction is a terminating or a recurring decimal; know when each happens, convert both ways by rule, and handle decimal points by counting places rather than guessing.

   Advanced · 6 min · Read here
7. 1.2.1.7Squares, square roots, cubes and cube roots: fast methods and estimation

   Factorise to find exact roots, use the last digit and the leading group to read roots of large perfect squares and cubes in seconds, and estimate the rest with a ± b/(2a).

   Advanced · 6 min · Read here
8. 1.2.1.8Surds and indices: laws, rationalising and comparing surds

   Bring every term to a common base to solve index questions, rationalise with the conjugate, and compare surds by raising them to a common power.

   Advanced · 5 min · Read here
9. 1.2.1.9Factorials: trailing zeros and the highest power of a prime in n!

   Count how many times a prime goes into n! by repeated division (n/p + n/p² + …), and every question on zeros, powers and divisibility of factorials becomes a short sum.

   Advanced · 6 min · Read here
10. 1.2.1.10Counting factors: number and sum of factors, perfect squares and co-primes

    Write N as a product of prime powers once; the number, sum and product of its factors, and every "how many factors are …" question, then follow by counting choices of exponents.

    Advanced · 6 min · Read here
11. 1.2.1.11Remainder theorems: cyclicity, Fermat, Wilson and negative remainders

    Replace each number by its remainder (or a small negative remainder), find a power that leaves 1 or −1, and the remainder of even a huge expression drops out in two lines.

    Advanced · 6 min · Read here
12. 1.2.1.12Problems on numbers: turning word statements into equations

    Name the unknown, translate each phrase into symbols, and solve one or two simple equations; in MCQs, checking the options against the conditions is often faster still.

    Advanced · 6 min · Read here
13. 1.2.1.13Speed maths: multiplication, squaring and percentage shortcuts

    Most aptitude arithmetic is a handful of patterns: turn 5, 25 and 125 into divisions, 9s into subtractions, nearby numbers into a base, and percentages into fractions.

    Advanced · 5 min · Read here
14. 1.2.1.14Place value, Roman numerals and counting digits in a range

    Read every digit by its place, write Roman numerals from the seven symbols and six subtractive pairs, and count digits place by place or block by block, never number by number.

    Advanced · 7 min · Read here

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