Infographics: Number System

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Number System 
The Number System forms the backbone of Quantitative Aptitude, comprising 15-20% of 
questions. This comprehensive visual guide presents essential tricks, formulas, and shortcuts 
across five core areas that frequently appear in the exam.
Number Properties
Master factorial zeroes, 
digital roots, and Euler's 
theorem for quick 
solutions
Factorisation
Calculate factors, their 
sums, products, and 
identify perfect squares 
efficiently
Divisibility
Apply instant divisibility 
tests and cyclicity 
patterns for rapid 
elimination
Remainders
Use modular arithmetic and theorem 
applications to crack complex problems
HCF & LCM
Leverage relationship formulas and 
fraction techniques for accurate results
Quick Formula Reference
Essential Summation Formulas
£n = n(n+1)/2
£n² = n(n+1)(2n+1)/6
£n³ = n²(n+1)²/4
Product of 3 consecutive numbers ÷ 6
x² - y² = (x+y)(x-y)
Key Relationships
A × B = HCF(A,B) × LCM(A,B)
Digital Root: Sum digits repeatedly
Factorial zeroes = Highest power of 5
Number of factors: (a+1)(b+1)(c+1)
Euler's: MÇ(N) ÷ N gives remainder 1
Factorisation Techniques
0 1
Prime Factorise
Express N = pa × qb × rc
0 2
Count Factors
Number of factors = (a+1)(b+1)(c+1)
0 3
Find Sum
Sum = [(pa+1-1)/(p-1)] × [(qb+1-1)/(q-1)]
0 4
Calculate Product
Product = N(x/2) where x = total factors
Divisibility Rules at a Glance
Divisibility by 2, 4, 8, 16
Check last 1, 2, 3, or 4 digits respectively
Divisibility by 3, 9, 27
For 3 & 9: sum of digits; For 27: sum of 3-
digit blocks
Divisibility by 7
Remove last digit, double it, subtract from 
truncated number
Divisibility by 11
(Sum of odd position digits) - (Sum of 
even position) = 0 or 11k
Worked Example: Factorisation
Problem: Find number of factors, sum, and product of factors fo r 1800
1
Step 1
1800 = 2³ × 3² × 5²
2
Step 2
Factors = (3+1)(2+1)(2+1) = 3 6
3
Step 3
Sum = 15 × 13 × 31 = 6 0 4 5
4
Step 4
Product = 180018
Remainder Shortcuts
Fermat's & Wilson's Theorems
Fermat's: Remainder of a(p-1) ÷ p = 1 (p is prime)
Wilson's: Remainder of (p-1)! ÷ p = (p-1) (p is prime)
Euler's Function: Ç(N) = N(1-1/a)(1-1/b)(1-1/c) for N = 
ap × bq × cr
Pro Tip for Multiplication
When finding remainder of products, replace each 
number with its remainder first, multiply, then find fin al 
remainder
HCF & LCM Applications
For Fractions
HCF = (HCF of 
numerators)/(LCM of 
denominators)
LCM = (LCM of 
numerators)/(HCF of 
denominators)
Remainder 
Problems
Greatest number 
dividing A, B, C with 
remainders p, q, r = 
HCF of (A-p), (B-q), (C-
r)
Same Remainder 
Case
Lowest number 
divisible by A, B, C 
leaving same remainder 
r = LCM(A,B,C) + r
Practice Problem: Even & Odd Factors
Question: Find the number of even and odd factors of 1200
Prime Factorise
1200 = 2t × 3¹ × 5²
Odd Factors
Exclude 2: (0+1)(1+1)(2+1) 
= 1 × 2 × 3 = 6
Even Factors
Include at least one 2: (4)
(2)(3) = 2 4
Critical Exam Tips
Speed Matters
Memorise cyclicity 
patterns of digits 2-9 and 
divisibility rules for 
instant application during 
the exam
Choose Wisely
Use remainders mod 
method for large number 
calculations instead of 
direct multiplication
Practice Daily
Solve 5-10 mixed 
problems daily covering 
all five topics to build 
pattern recognition
Key Takeaways
Master the 
Formulas
Commit factorisation 
formulas, sum series, 
and theorem 
applications to memory 
for 30-second problem 
solving
Understand Co-
primality
Euler's theorem and 
remainder concepts 
become trivial when 
you identify co-prime 
relationships quickly
Leverage 
Properties
Use number properties 
like divisibility of 
consecutive products 
(by 6, by 24) to 
eliminate options 
rapidly