Number System & Simplification — Quantitative Aptitude Notes for IOCL Graduate Engineer Civil

Part 1 of 2

Number System

Last reviewed 16 Sept 2026 · 8 min read

Type	Definition	Examples
Natural numbers (N)	Counting numbers	1, 2, 3, …
Whole numbers (W)	Natural numbers and zero	0, 1, 2, …
Integers (Z)	Whole numbers and negatives	…, −2, −1, 0, 1, 2, …
Rational numbers (Q)	Expressible as $p / q$ ( $q \neq = 0$ ); terminating or recurring decimals	3/4, −5, 0.333…
Irrational numbers	Non-terminating, non-recurring decimals	√2, π, √3
Real numbers (R)	Rational + irrational	All of the above
Even numbers	Divisible by 2	0, 2, 4
Odd numbers	Not divisible by 2	1, 3, 5
Prime numbers	Exactly two factors (1 and itself)	2, 3, 5, 7, 11
Composite numbers	More than two factors	4, 6, 8, 9
Co-prime numbers	HCF = 1	(8, 15), (9, 16)

Divisor	Rule	Example
2	Last digit even	3 458 ✓
3	Sum of digits divisible by 3	7 248: 7 + 2 + 4 + 8 = 21 ✓
4	Last two digits divisible by 4	13 24 ✓
5	Last digit 0 or 5	985 ✓
6	Divisible by both 2 and 3	1 254 ✓
7	Double the last digit and subtract from the rest; repeat	343: 34 − 6 = 28 ✓
8	Last three digits divisible by 8	7 248: 248/8 = 31 ✓
9	Sum of digits divisible by 9	5 832: 18 ✓
10	Last digit 0	4 560 ✓
11	Difference of sums of alternate digits is 0 or divisible by 11	918 082: (9 + 8 + 8) − (1 + 0 + 2) = 22 ✓
12	Divisible by 3 and 4	1 332 ✓
13	Add 4 × last digit to the rest; repeat	169: 16 + 36 = 52 = 4 × 13 ✓

For composite divisors use co-prime factors (e.g. 18 = 2 × 9, 24 = 3 × 8, 36 = 4 × 9, 72 = 8 × 9).

Unit digits of powers repeat in cycles:

Method: divide the power by 4; use the remainder (remainder 0 → take the 4th term of the cycle).

FormulaFactors of a number

If $N = a^{p} \times b^{q} \times c^{r}$ (prime factorisation):

Number of factors $= (p + 1) (q + 1) (r + 1)$
Sum of factors $= \frac{a ^{p + 1} - 1}{a - 1} \times \frac{b ^{q + 1} - 1}{b - 1} \times \frac{c ^{r + 1} - 1}{c - 1}$
Product of factors $= N^{(number of factors) /2}$
Number of odd factors — ignore the power of 2: $(q + 1) (r + 1)$ when $a = 2$
A number has an odd number of factors only if it is a perfect square

FormulaRemainder rules

Part 2 of 2

Last reviewed 16 Sept 2026 · 6 min read

Letter	Operation	Order
V	Vinculum (bar over numbers — e.g. $\overline{8 - 3}$ )	First
B	Brackets — in order: ( ) → { } → [ ] (innermost first)	Second
O	Of (means multiplication, e.g. ½ of 20) and orders/powers/roots	Third
D	Division	Fourth (D and M from left to right)
M	Multiplication
A	Addition	Last (A and S from left to right)
S	Subtraction

Key point: Division and multiplication have equal priority — evaluate left to right; similarly addition and subtraction.

FormulaUseful identities

$(a + b)^{2} = a^{2} + 2 ab + b^{2}$ ; $(a - b)^{2} = a^{2} - 2 ab + b^{2}$
$a^{2} - b^{2} = (a + b) (a - b)$
$(a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})$ ; $(a + b)^{2} - (a - b)^{2} = 4 ab$
$(a + b)^{3} = a^{3} + b^{3} + 3 ab (a + b)$ ; $(a - b)^{3} = a^{3} - b^{3} - 3 ab (a - b)$
$a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ ; $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$
$a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)$ ; if $a + b + c = 0$ , then $a^{3} + b^{3} + c^{3} = 3 ab c$
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + b c + c a)$

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