SBI PO Mains – Number System (Complete)
Types of Numbers
Special Numbers
Divisibility
HCF–LCM
Unit Digit
Remainder
Trailing Zeroes
Highest Power
Factors
Sum of Divisors
Perfect Sq/Cube
Sum Formulas
Number Formation
Digital Root
Even–Odd Rules
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1. Classification of Numbers
Real Numbers: Can be plotted on a number line.Imaginary Numbers: Cannot be plotted (e.g., i = √-1 ).
Rational Numbers: Can be written as p/q (q≠0).Irrational Numbers: Non-terminating, non-recurring (e.g., √2, π).
Recurring Decimal to Fraction: 2.666... = 24/9 = 8/3.
Integers: Negative, Zero (neutral), Positive.Whole Numbers: Natural numbers including Zero.Natural Numbers: Counting numbers (1, 2, 3...).
Primes: Exactly 2 factors. 2 is the only even prime. 1 is neither prime nor composite.
2. More About Numbers
6k ± 1 Rule: Every prime > 3 is of this form.Co-prime: HCF = 1. E.g., (7, 11).Consecutive Product: Product of 3 consecutive numbers is divisible by 6.Sum of two primes is even unless one is 2.
3. Surds & Indices
am × an = am+n am ÷ an = am-n (am )n = amn a-n = 1/an a0 = 1
Rationalization: 1/(√2+1) = √2-1 (multiply by conjugate).
4. Test of Divisibility
Divisor Rule 2 Last digit even 3 Sum of digits div by 3 4 Last 2 digits div by 4 5 Last digit 0 or 5 8 Last 3 digits div by 8 9 Sum of digits div by 9 11 (Sum odd places) - (Sum even places) = 0, 11
Composite: 12 (3&4), 72 (8&9), 80 (16&5), 125 (Last 3 digits).
5. Remainder Theorem
Formula: Dividend = (Divisor × Quotient) + Remainder .
Rule 1: Add/Subtract remainders.Rule 2: Multiply remainders.Negative Remainder: Rem 7 mod 9 = -2 mod 9.
Cyclicity: 2243 /9. Cycle length 6. 243/6 = rem 3. 23 = 8.
6. HCF & LCM
Formula: HCF × LCM = Product of two numbers .
Fractions: LCM = LCM(num)/HCF(den), HCF = HCF(num)/LCM(den).
Word Problems: Greatest number dividing a,b,c leaving remainders x,y,z: HCF(a-x, b-y, c-z). Least number divided by a,b,c leaving remainders x,y,z: LCM(a,b,c) - (a-x) or (b-y) or (c-z).
7. Unit Digit, Powers & Factorials
Number Cycle Pattern 2 2, 4, 8, 6 3 3, 9, 7, 1 7 7, 9, 3, 1 8 8, 4, 2, 6 4 & 9 4, 6 & 9, 1
Trailing Zeros in 100!: [100/5] + [100/25] = 24.
Max Power of 7 in 67!: [67/7] + [67/49] = 10.
8. Misc & Advanced
Rule of 1001: abcabc divisible by 7, 11, 13.Polydivisible: 381654729.Nested Quotient: N/13 rem 1, Q/5 rem 3 ⇒ N = 65y + 40 ⇒ Rem 40.
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