Number System
Why This Chapter Matters
Number System forms the base of all quantitative aptitude — tested in SBI PO, IBPS PO, IBPS Clerk, RRB every year. Strong fundamentals here make all other topics easier.
Core Concepts
1. Types of Numbers
Natural numbers: 1,2,3,... | Whole: 0,1,2,... | Integers: ...,-2,-1,0,1,2,...
Rational: p/q form | Irrational: sqrt(2), pi | Real: all of the above
Even: divisible by 2. Odd: not divisible by 2.
Prime: divisible only by 1 and itself (2,3,5,7,11,13...)
Composite: more than 2 factors
Special: 1 is neither prime nor composite. 2 is only even prime.
2. Divisibility Rules
Div by 2: last digit even. Div by 3: digit sum divisible by 3. Div by 4: last 2 digits div by 4.
Div by 5: ends in 0 or 5. Div by 6: div by both 2 and 3. Div by 8: last 3 digits div by 8.
Div by 9: digit sum div by 9. Div by 10: ends in 0. Div by 11: alternating digit sum divisible by 11.
Div by 11 trick: (sum of digits at odd places) - (sum at even places) = 0 or 11.
3. HCF and LCM
HCF (GCD): largest number dividing both. HCF(a,b) using Euclid: a = bq + r, then HCF(b,r).
LCM: smallest number divisible by both.
HCF x LCM = Product of two numbers (for EXACTLY TWO numbers).
4. Important Number Properties
Consecutive integers: (n)(n+1) always even. n(n+1)(n+2) always divisible by 6.
Sum of first n natural numbers: n(n+1)/2.
Sum of first n odd numbers: n^2.
Sum of first n even numbers: n(n+1).
5. Surds and Indices
a^m x a^n = a^(m+n). a^m / a^n = a^(m-n). (a^m)^n = a^(mn).
a^0 = 1 (any non-zero a). a^(-n) = 1/a^n.
sqrt(a) x sqrt(b) = sqrt(ab). n-th root of a = a^(1/n).
Practice Problems
Q1: HCF of 96 and 404?
96 = 2^5 x 3. 404 = 2^2 x 101. HCF = 2^2 = 4.
Q2: Find LCM of 12, 18, 24.
12=2^2x3, 18=2x3^2, 24=2^3x3. LCM = 2^3 x 3^2 = 72.
Q3: Which is largest: 2^32 or 8^12?
8^12 = (2^3)^12 = 2^36. So 2^32 < 2^36. 8^12 is larger.
Q4: If a number is divisible by both 4 and 9, is it divisible by 36?
LCM(4,9)=36 (since HCF=1, they are coprime). Yes, divisible by 36.
Previous Year Questions
SBI PO 2023: Find the largest 4-digit number divisible by 88.
9999/88 = 113.6... So 113 x 88 = 9944. Answer: 9944.
IBPS PO 2022: What is the sum of all prime numbers between 20 and 30?
23 and 29. Sum = 52.
RRB 2021: If LCM of two numbers is 96 and HCF is 4, one number is 12. Find the other.
Other = (LCM x HCF)/first = (96 x 4)/12 = 32.