Digit = one symbol
The decimal system uses exactly ten digits:
\[\{0,1,2,3,4,5,6,7,8,9\}\]
Digits are the alphabet from which numerals are written.
1 SSC CGL · Mathematics
Number System & Basic Computation
Complete handwritten-style notes for SSC CGL Tier I and Tier II.
01
MAP IT!
The Big Picture — Graphic Mind Map
Every number-system question is one of six jobs: identify, compare, divide, simplify, cycle or estimate.
NUMBER
SYSTEM
SYSTEM
1. Number FamiliesN, W, Z, Q, irrational and real numbers
2. Factors & Multiplesprime, composite, HCF, LCM and divisibility
3. Fractions & Decimalscompare, operate, convert and classify
4. SimplificationBODMAS, signs, indices, roots and approximation
5. Remaindersmodular reduction, divisibility and cyclic powers
6. Speed Arithmeticunit digit, digit sum, cancellation and estimation
Exam lens: classify the number ➜ identify the property ➜ choose the shortest operation ➜ check the option.
02
FOUNDATION
The Foundation (Basics) — Core Concepts
Start from digit, number and place value; then build every rule one small idea at a time.
0Before everything: digit, numeral and number
Numeral = written form
\(507\) is a numeral made from the digits \(5,0,7\). Its position tells each digit's value.
\[507=5\times100+0\times10+7\times1\]
Number = the quantity
A number answers “how many?”, “which position?” or “how much?”. Different numerals can name the same number:
\[\frac12=0.5=50\%\]
Base-10 place-value rule: \[(a_na_{n-1}\ldots a_1a_0)_{10}=\sum_{k=0}^{n}a_k10^k,\qquad 0\le a_k\le9\]
See place value like three labelled classrooms
A digit carries its face value into a position. The position supplies the multiplier. In the numeral \(507\), the zero is important: it keeps the tens place empty without shifting the other digits.
symbolpositionvalue
The zero holds the middle seat.
AThe number family
Nested Sets — who lives inside whom?
Real numbers \(\mathbb R\): rational and irrational
Rational numbers \(\mathbb Q\): \(\frac pq\), where \(q\ne0\)
Integers \(\mathbb Z=\{\ldots,-2,-1,0,1,2,\ldots\}\)
Whole numbers \(\mathbb W=\{0,1,2,3,\ldots\}\)
Natural numbers \(\mathbb N=\{1,2,3,\ldots\}\)
\[\mathbb{N}\subset\mathbb{W}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\]
Irrational numbers also belong to \(\mathbb{R}\) but not \(\mathbb{Q}\): \(\sqrt2,\sqrt3,\pi\) and non-terminating non-repeating decimals.
Quick identity cards
- Natural: counting numbers; SSC convention normally begins at 1.
- Whole: natural numbers plus 0.
- Integer: negative whole numbers, 0 and positive whole numbers.
- Rational: can be written as \(\frac pq\), with \(p,q\in\mathbb Z\) and \(q\ne0\).
- Irrational: decimal neither terminates nor repeats.
- Real: every point on the number line.
Key: terminating or repeating decimal = rational.
BSigns, order and number line
Number-line law
Move right ➜ the number becomes larger. Move left ➜ it becomes smaller.
−4−2024
\(-7\lt-3\) because \(-7\) lies farther left.
Sign rules
\[(+)\times(+)=(-)\times(-)=+\qquad\text{and}\qquad (+)\times(-)=-\]
- Adding same signs: add magnitudes, keep the sign.
- Adding unlike signs: subtract magnitudes, keep the sign of the larger magnitude.
- \(|a|\) is distance from \(0\), so \(|a|\ge0\).
- An even integer is divisible by \(2\); \(0\) is even.
- \(\text{odd}\pm\text{odd}=\text{even}\); \(\text{odd}\times\text{odd}=\text{odd}\).
CPlace value, face value and base-10 form
Face Value
The digit itself, wherever it sits.
\[\operatorname{face\ value}_{57284}(7)=7\]
Place Value
Digit multiplied by the value of its position.
\[\operatorname{place\ value}_{57284}(7)=7\times10^3=7000\]
Expanded Form
\[57284=5\times10^4+7\times10^3+2\times10^2+8\times10+4\]
To the right of the decimal: tenths \((10^{-1})\), hundredths \((10^{-2})\), thousandths \((10^{-3})\), and so on.
DFactors, multiples, prime and composite numbers
Factor
A number that divides exactly.
\[\operatorname{Factors}(12)=\{1,2,3,4,6,12\}\]
Finite list. Positive factors of \(n\) do not exceed \(n\).
Multiple
Product of the number with an integer.
\[\operatorname{Multiples}(12)=\{12,24,36,48,\ldots\}\]
Infinite list.
Prime vs Composite
Prime: exactly two positive factors, \(1\) and itself.
Composite: more than two positive factors.
\[1\text{ is neither prime nor composite.}\]
Fundamental Theorem of Arithmetic: every integer \(n\gt1\) has a unique prime factorization apart from the order of factors.
Factors branch inward; multiples march outward
A factor tree breaks one number into prime building blocks. A multiples chain keeps adding the original number forever.
\[60=2^2\times3\times5\qquad 12n:\ 12,24,36,\ldots\]
EDivisibility rules — the no-division toolkit
| Divisor | Fast Test | Example |
|---|---|---|
| \(2\) | Last digit is \(0,2,4,6\) or \(8\). | \(4738\) ✓ |
| \(3\) | Sum of digits is divisible by \(3\). | \(7452\to7+4+5+2=18\) ✓ |
| \(4\) | Last two digits form a multiple of \(4\). | \(31716\to16=4\times4\) ✓ |
| \(5\) | Last digit is \(0\) or \(5\). | \(8235\) ✓ |
| \(6\) | Number is divisible by both \(2\) and \(3\). | \(5184\) is even and its digit sum is \(18\) ✓ |
| \(7\) | Remove the last digit, double it and subtract from the remaining number; repeat. | \(203\to20-2(3)=14\) ✓ |
| \(8\) | Last three digits form a multiple of \(8\). | \(47216\to216=27\times8\) ✓ |
| \(9\) | Sum of digits is divisible by \(9\). | \(83736\to8+3+7+3+6=27\) ✓ |
| \(10\) | Last digit is \(0\). | \(9430\) ✓ |
| \(11\) | Difference of alternating digit sums is \(0\) or a multiple of \(11\). | \(4719:(4+1)-(7+9)=-11\) ✓ |
| \(12\) | Number is divisible by both \(3\) and \(4\). | \(4356\): digit sum \(18\), and \(56=14\times4\) ✓ |
| \(13\) | Remove the last digit, multiply it by \(4\), then add to the remaining number; repeat. | \(286\to28+4(6)=52\) ✓ |
| \(15\) | Number is divisible by both \(3\) and \(5\). | \(2745\): digit sum \(18\), ends in \(5\) ✓ |
| \(16\) | Last four digits form a multiple of \(16\). | \(15312\to5312=332\times16\) ✓ |
| \(25\) | Last two digits are \(00,25,50\) or \(75\). | \(18575\) ✓ |
| \(125\) | Last three digits are divisible by \(125\). | \(987250\to250=2\times125\) ✓ |
The divisibility scanner asks only three questions
- Last digits? Use for powers of \(2\) and \(5\).
- Digit sum? Use for \(3\) and \(9\).
- Alternating sum? Use for \(11\).
The entire number rarely needs long division.
Pick the correct scanner before calculating.
FHCF and LCM — common factor vs common schedule
HCF / GCD
Highest common factor dividing every given number.
Use for: greatest possible size, maximum equal groups, largest tile/measure.
\[v_p(\operatorname{HCF})=\min\{v_p(a),v_p(b),\ldots\}\]
Example: \(72=2^3\times3^2\), \(120=2^3\times3\times5\), so \(\operatorname{HCF}=2^3\times3=24\).
LCM
Least common multiple divisible by every given number.
Use for: earliest coincidence, repeating bells/lights, smallest common quantity.
\[v_p(\operatorname{LCM})=\max\{v_p(a),v_p(b),\ldots\}\]
For the same pair, \(\operatorname{LCM}=2^3\times3^2\times5=360\).
\[\text{For two positive integers }a,b:\quad \operatorname{HCF}(a,b)\operatorname{LCM}(a,b)=ab\]
Meaning before method
HCF cuts resources into the greatest identical groups. LCM finds the earliest time repeating events meet again.
greatest sizeequal groupsfirst coincidence
GFractions — compare, operate and convert
Types
- Proper: \(|a|\lt|b|\) for \(\frac ab\).
- Improper: \(|a|\ge|b|\).
- Mixed: integer plus a proper fraction.
- Equivalent: \(\frac ab=\frac{ka}{kb}\) for \(k\ne0\).
Operations
\[\frac ab\pm\frac cd=\frac{ad\pm bc}{bd}\]
\[\frac ab\times\frac cd=\frac{ac}{bd}\]
\[\frac ab\div\frac cd=\frac ab\times\frac dc\quad(c\ne0)\]
Cancel common factors before multiplying.
Comparison
For \(b,d\gt0\), compare cross-products:
\[\frac ab\ ?\ \frac cd\quad\Longleftrightarrow\quad ad\ ?\ bc\]
If denominators are equal, the larger numerator wins. If positive numerators are equal, the smaller denominator wins.
HDecimals — terminating or recurring?
Fraction ⇄ Decimal
- Fraction to decimal: compute \(p\div q\).
- For a terminating decimal with \(k\) places, remove the decimal and divide by \(10^k\); then reduce.
\[0.375=\frac{375}{1000}=\frac38\]
\[0.\overline{27}=\frac{27}{99}=\frac3{11}\]
Termination test
Reduce \(\frac pq\) to lowest terms. Its decimal terminates only if \(q\) has no prime factors other than \(2\) and \(5\).
\[q=2^m5^n\quad\Longleftrightarrow\quad\frac pq\text{ has a terminating decimal}\]
Otherwise it is non-terminating recurring, hence rational.
The decimal gate has only two keys
After cancellation, imagine the denominator approaching a gate guarded by prime keys \(2\) and \(5\). If any other prime factor remains, the decimal repeats.
\[\frac pq\text{ terminates}\iff q=2^m5^n\quad(m,n\in\mathbb W)\]
IBODMAS, powers and roots
BBrackets
OOrders / powers / roots
D & MLeft to right
A & SLeft to right
CheckEstimate sign and size
Indices
\[a^ma^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n}\ (a\ne0)\]
\[(a^m)^n=a^{mn},\qquad a^0=1\ (a\ne0)\]
\[a^{-n}=\frac1{a^n},\qquad (ab)^n=a^nb^n\]
Roots and perfect powers
\[\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)\]
A perfect square has even prime exponents. A perfect cube has prime exponents divisible by \(3\).
\[n=\prod p_i^{\alpha_i}:\quad n\text{ square}\iff2\mid\alpha_i;\quad n\text{ cube}\iff3\mid\alpha_i\]
Order is a staircase, not a race
Complete the higher step first. Equal-priority twins—division with multiplication, addition with subtraction—walk from left to right.
bracketsordersleft to right
JRemainders, unit digits and approximation
Division algorithm
\[N=dq+r,\qquad 0\le r\lt d\quad(d\gt0)\]
Here \(N\) is dividend, \(d\) divisor, \(q\) quotient and \(r\) remainder.
Remainder reduction
\[(a+b)\bmod m=\big((a\bmod m)+(b\bmod m)\big)\bmod m\]
\[(ab)\bmod m=\big((a\bmod m)(b\bmod m)\big)\bmod m\]
Approximation
Round only to the requested place. Estimate sign, order of magnitude and last digit before exact work.
\[x=d_0.d_1d_2\ldots;\quad\text{next digit}\ge5\Rightarrow\text{round up}\]
Remainder is the seat on a circular clock
Moving by a full cycle changes the quotient but returns to the same remainder. Reduce large powers and products at each step; never carry unnecessary size.
\[a\equiv b\pmod m\Longrightarrow a^k\equiv b^k\pmod m\]
A full lap adds no new remainder.
KProperties of operations — what is always safe?
| Property | Addition | Multiplication | Warning |
|---|---|---|---|
| Closure | \(a+b\) stays in \(\mathbb Z,\mathbb Q,\mathbb R\). | \(ab\) stays in \(\mathbb Z,\mathbb Q,\mathbb R\). | \(a/b\) need not be an integer; division by \(0\) is undefined. |
| Commutative | \(a+b=b+a\) | \(ab=ba\) | Subtraction and division are not commutative. |
| Associative | \((a+b)+c=a+(b+c)\) | \((ab)c=a(bc)\) | Subtraction and division are not associative. |
| Identity | \(a+0=a\) | \(a\cdot1=a\) | \(0\) is additive identity; \(1\) is multiplicative identity. |
| Inverse | \(a+(-a)=0\) | \(a\cdot a^{-1}=1\), \(a\ne0\) | \(0\) has no multiplicative inverse. |
| Distributive | \(a(b+c)=ab+ac\) and \(a(b-c)=ab-ac\) | This links multiplication with addition/subtraction. | |
Rational–irrational outcomes
- \(\text{rational}+\text{rational}=\text{rational}\).
- \(\text{rational}+\text{irrational}=\text{irrational}\).
- A non-zero rational times an irrational is irrational.
- Two irrationals may sum/product to rational or irrational: \(\sqrt2+(-\sqrt2)=0\), but \(\sqrt2+\sqrt3\) is irrational.
Density property
Between any two distinct real numbers lie infinitely many rational and infinitely many irrational numbers.
\[a\lt b\quad\Longrightarrow\quad a\lt\frac{a+b}{2}\lt b\]
The midpoint gives one rational when \(a,b\) are rational; repeat to generate infinitely many.
LFactor-count and divisor formula mastery
Number of divisors
If \(n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}\), then
\[\tau(n)=\prod_{i=1}^{k}(a_i+1)\]
For \(72=2^3\cdot3^2\): \(\tau(72)=(3+1)(2+1)=12\).
Sum and product of divisors
\[\sigma(n)=\prod_{i=1}^{k}\frac{p_i^{a_i+1}-1}{p_i-1}\]
\[\prod_{d\mid n}d=n^{\tau(n)/2}\]
Sum of proper divisors is \(\sigma(n)-n\).
Odd/even divisor count
Write \(n=2^a m\), where \(m\) is odd.
\[\#\text{ odd divisors}=\tau(m)\]
\[\#\text{ even divisors}=\tau(n)-\tau(m)\]
Prime test
To test whether \(n\gt1\) is prime, check divisibility only by primes \(p\le\sqrt n\). If no such prime divides \(n\), then \(n\) is prime.
Co-prime numbers
\(a,b\) are co-prime iff \(\gcd(a,b)=1\). They need not be prime: \(8\) and \(15\) are co-prime.
\[\gcd(a,b)=1\Rightarrow\operatorname{LCM}(a,b)=ab\]
Perfect-number link
A positive integer is perfect when its proper divisors sum to itself.
\[\sigma(n)=2n\]
Examples: \(6=1+2+3\), \(28=1+2+4+7+14\).
Divisors arrive in pairs
Each factor below \(\sqrt n\) pairs with one above \(\sqrt n\). A perfect square has one central self-pair, which is why its divisor count is odd.
\[d\mid n\Longrightarrow d\cdot\frac nd=n\]
MHCF–LCM variants and Euclid's algorithm
Euclidean algorithm
\[\gcd(a,b)=\gcd\bigl(b,a\bmod b\bigr)\]
Repeat until the remainder becomes \(0\); the last non-zero remainder is the HCF.
\(\gcd(252,105)=\gcd(105,42)=\gcd(42,21)=21\).
Fractions in lowest terms
\[\operatorname{HCF}\left(\frac ab,\frac cd\right)=\frac{\gcd(a,c)}{\operatorname{LCM}(b,d)}\]
\[\operatorname{LCM}\left(\frac ab,\frac cd\right)=\frac{\operatorname{LCM}(a,c)}{\gcd(b,d)}\]
Co-prime core form
If \(a=dx\), \(b=dy\) and \(\gcd(x,y)=1\), then
\[\gcd(a,b)=d,\qquad\operatorname{LCM}(a,b)=dxy\]
NRemainder theorems and edge cases
Congruence language
\[a\equiv b\pmod m\iff m\mid(a-b)\]
Numbers congruent modulo \(m\) leave the same remainder on division by \(m\).
Same-remainder theorem
If a divisor \(d\) leaves the same remainder when dividing \(N_1,N_2,\ldots\), then \(d\) divides every pairwise difference.
\[d\mid(N_i-N_j)\]
The greatest possible \(d\) is the HCF of the differences.
Specified remainders
If \(N_i\) leaves remainder \(r_i\) on division by \(d\), then
\[d\mid(N_i-r_i)\]
Find the HCF of adjusted numbers, then verify \(d\gt r_i\).
Negative remainder repair
If calculation gives \(-r\pmod m\), convert to the least non-negative remainder.
\[-r\equiv m-r\pmod m\quad(0\lt r\lt m)\]
Fermat's little theorem
For prime \(p\) with \(p\nmid a\):
\[a^{p-1}\equiv1\pmod p\]
Equivalent form: \(a^p\equiv a\pmod p\).
Euler's theorem
When \(\gcd(a,n)=1\):
\[a^{\varphi(n)}\equiv1\pmod n\]
For \(n=\prod p_i^{a_i}\), \(\varphi(n)=n\prod_{p\mid n}(1-1/p)\).
OPerfect squares, cubes and digit facts
Square fingerprints
- A square cannot end in \(2,3,7,8\).
- Trailing zeros in a non-zero perfect square occur in an even count.
- Between \(n^2\) and \((n+1)^2\) lie exactly \(2n\) integers.
- The number of odd integers from \(1\) through \(2n-1\) sums to \(n^2\).
\[1+3+5+\cdots+(2n-1)=n^2\]
Cube fingerprints
Each possible unit digit of a cube identifies the unit digit of the base.
\[0\to0,\ 1\to1,\ 2\to8,\ 3\to7,\ 4\to4,\ 5\to5,\ 6\to6,\ 7\to3,\ 8\to2,\ 9\to9\]
Trailing zeros in a non-zero perfect cube occur in multiples of \(3\).
How many digits?
\[\#\text{digits of }N=\lfloor\log_{10}N\rfloor+1\quad(N\ge1)\]
\[\#\text{positive }n\text{-digit integers}=9\times10^{n-1}\]
Smallest \(n\)-digit number is \(10^{n-1}\); largest is \(10^n-1\).
Square means paired layers; cube means tripled layers
Prime exponents are construction instructions. Pair every layer for a square; group every three layers for a cube. Unmatched layers show exactly what must be multiplied or divided.
pairs → squaretriples → cubeunmatched → repair
PMust-know data banks
| Dataset | Values to Memorize |
|---|---|
| Primes below \(100\) | \(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97\) |
| Squares \(1^2\) to \(15^2\) | \(1,4,9,16,25,36,49,64,81,100,121,144,169,196,225\) |
| Squares \(16^2\) to \(30^2\) | \(256,289,324,361,400,441,484,529,576,625,676,729,784,841,900\) |
| Cubes \(1^3\) to \(10^3\) | \(1,8,27,64,125,216,343,512,729,1000\) |
| Cubes \(11^3\) to \(20^3\) | \(1331,1728,2197,2744,3375,4096,4913,5832,6859,8000\) |
| Common fractions | \(\frac12=0.5,\ \frac13=0.\overline3,\ \frac14=0.25,\ \frac15=0.2,\ \frac18=0.125,\ \frac1{11}=0.\overline{09}\) |
03
SAVE TIME
Short Tricks & Magic Formulas — Ninja Techniques
Choose shortcuts that preserve logic; never memorize a trick without its condition.
1HCF–LCM in one line
Find the missing LCM
If \(\operatorname{HCF}(84,126)=42\):
\[\operatorname{LCM}(84,126)=\frac{84\times126}{42}=252\]
Valid because there are exactly two positive integers.
Prime-exponent camera
Write each number in primes. HCF takes the low power; LCM takes the large power.
\[v_p(\operatorname{HCF})=\min(v_p),\qquad v_p(\operatorname{LCM})=\max(v_p)\]
2Divisibility six-pack
\(2\)last \(1\) digit even
\(3\)digit sum divisible by \(3\)
\(4\)last \(2\) digits
\(8\)last \(3\) digits
\(9\)digit sum divisible by \(9\)
\(11\)alternate-sum test
Composite shortcut: \[6=2\times3,\quad12=3\times4,\quad15=3\times5,\quad18=2\times9\] Test the coprime parts.
3Unit-digit cycles — powers without expansion
Cycle chart
Only the base's unit digit matters.
\(2^n:\)\(2\)\(4\)\(8\)\(6\)repeat
\(3^n:\)\(3\)\(9\)\(7\)\(1\)repeat
\(7^n:\)\(7\)\(9\)\(3\)\(1\)repeat
\(8^n:\)\(8\)\(4\)\(2\)\(6\)repeat
Example: unit digit of \(7^{103}\)
- Cycle length of \(7^n\) is \(4\).
- \(103=4(25)+3\), so the exponent position is \(3\).
- Take the third cycle entry: \(3\).
\[7^{103}\equiv3\pmod{10}\]
If the exponent leaves remainder \(0\) on division by cycle length, use the cycle's last entry.
4Butterfly fraction comparison
Compare \(\frac7{12}\) and \(\frac58\)
\(\frac7{12}\)\(\frac58\)
\(7\times8=56\) and \(5\times12=60\)
\[56\lt60\quad\Longrightarrow\quad\frac7{12}\lt\frac58\]
Near-\(1\) shortcut
For positive proper fractions, compare what is missing from \(1\).
\[1-\frac{11}{12}=\frac1{12},\qquad1-\frac{14}{15}=\frac1{15}\]
Smaller deficiency means larger fraction, so \(\frac{14}{15}\gt\frac{11}{12}\).
5Recurring-decimal flash conversion
One repeating digit
\[0.\overline7=\frac79\]
Two repeating digits
\[0.\overline{36}=\frac{36}{99}=\frac4{11}\]
Three repeating digits
\[0.\overline{125}=\frac{125}{999}\]
Pure recurring master rule
If a block of \(s\) digits repeats immediately after the decimal, use \(s\) nines.
\[0.\overline{A}=\frac{A}{10^s-1}\]
Mixed recurring: all minus non-repeat
Write one full repeating block, subtract the non-repeating part, then place \(s\) nines followed by \(r\) zeros below.
\[0.12\overline3=\frac{123-12}{900}=\frac{37}{300}\]
Visual cue: repeating digits create \(9\)s; non-repeating digits create trailing \(0\)s in the denominator.
6Remainder and digit-sum hacks
Reduce before multiplying
Remainder of \(87\times94\) on division by \(7\):
\[87\equiv3,\quad94\equiv3\pmod7\quad\Longrightarrow\quad87\cdot94\equiv9\equiv2\pmod7\]
Digital-root check
Digit sum preserves remainder modulo \(9\).
\[58734\equiv5+8+7+3+4=27\equiv0\pmod9\]
Use as a check, not as a complete proof for every divisor.
Trailing zero count
Each zero needs a \(2\times5\) pair; factorials contain more \(2\)s, so count \(5\)s.
\[Z(n!)=\left\lfloor\frac n5\right\rfloor+\left\lfloor\frac n{25}\right\rfloor+\left\lfloor\frac n{125}\right\rfloor+\cdots\]
7Formula mastery board — edge-case speed tools
Count multiples in an interval
\[\#\{d\text{-multiples in }[L,R]\}=\left\lfloor\frac Rd\right\rfloor-\left\lfloor\frac{L-1}{d}\right\rfloor\]
Prime power inside a factorial
\[v_p(n!)=\sum_{k\ge1}\left\lfloor\frac{n}{p^k}\right\rfloor\]
Stop when \(p^k\gt n\).
Least adjustment to divisibility
If \(N=dq+r\), subtract \(r\), or add the complement to the next multiple.
\[\text{least add}=(d-r)\bmod d,\qquad\text{least subtract}=r\]
Last \(k\) digits
Work modulo the matching power of ten.
\[\text{last }k\text{ digits of }N=N\bmod10^k\]
Terminating decimal length
For reduced denominator \(q=2^m5^n\), the number of decimal places is
\[\max(m,n)\]
Consecutive-integer fact
\[\gcd(n,n+1)=1\]
Any two consecutive integers are co-prime; one of them is even.
Formula dashboard
Do not start expanding a huge number. Ask what the options actually need: a count, a remainder, a last digit, or only divisibility.
reducecountadjustverify
8Option-first speed checks
±check sign
≈estimate size
Ucheck unit digit
9digit-sum check
×cancel first
Rtest remainder
04
DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
These are option-setters' favourite places to punish automatic calculation.
Trap 1: 0 and 1 are special
- \(0\) is whole, integer, rational, real and even.
- \(0\) is neither positive nor negative.
- \(1\) is neither prime nor composite.
- The smallest prime is \(2\), the only even prime.
Trap 2: BODMAS twins move left-to-right
Division does not automatically defeat an earlier multiplication; multiplication and division have equal priority.
\[24\div6\times2=4\times2=8\ne2\]
Likewise, addition and subtraction move left to right.
Trap 3: formula condition hidden in language
- \(\operatorname{HCF}\times\operatorname{LCM}=\text{product}\) applies directly to two positive integers, not any three numbers.
- A remainder satisfies \(0\le r\lt d\) for positive divisor \(d\).
- Exponent-cycle remainder \(0\) means use the cycle's last position.
- Reduce \(\frac pq\) before applying the terminating-decimal test.
05
LOCK IT IN
Memory Hooks & Mnemonics
Attach each hard rule to a picture, sentence or rhythm that your brain can retrieve under pressure.
“Natural Wolves Zigzag Quietly to Reality”
\(\mathbb N\subset\mathbb W\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\). Irrational numbers live in Reality \((\mathbb R)\) but outside the Quiet rational room \((\mathbb Q)\).
“Four sees Two; Eight sees Three”
For divisibility by \(4\), inspect the last \(2\) digits. For divisibility by \(8\), inspect the last \(3\) digits.
“Factors Fit; Multiples March”
Factors fit exactly inside the number and stop. Multiples march outward forever.
“HCF goes LOW; LCM goes LARGE”
For every prime \(p\), HCF takes the minimum exponent \(v_p\) and LCM takes the maximum.
“Only 2 and 5 can terminate the call”
After reducing \(\frac pq\), a denominator \(2^m5^n\) gives a terminating decimal; any other prime factor forces recurrence.
“D-M twins, A-S twins: walk left to right”
Division and multiplication share one level; addition and subtraction share one level. Position decides.
PrimeExactly \(2\) positive factors
Co-prime\(\operatorname{HCF}=1\); numbers need not be prime
Perfect squareAll prime exponents even
HCFGreatest equal division
LCMEarliest common repetition
Remainder\(0\le r\lt d\) for positive divisor \(d\)