Number System

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Number System — Quant Aptitude Questions & Formulas

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39 questions

1. Divisibility Rules

Divisor	Rule
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit is 0
11	(Sum of odd-place digits) − (sum of even-place digits) is 0 or a multiple of 11
7	Drop last digit d, take rest + 5d; repeat
13	Drop last digit d, take rest + 4d; repeat
17	Drop last digit d, take rest − 5d; repeat
19	Drop last digit d, take rest + 2d; repeat

2. HCF & LCM

For two numbers, $\text{HCF} \times \text{LCM} = a \times b$ , so $\text{LCM} = \dfrac{a\times b}{\text{HCF}}$ and .

Fractions: $\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}$ and $\text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$ .

3. Prime Factorisation & Factors

If $N = p^{a}\,q^{b}\,r^{c}$ with $p, q, r$ prime:

Total factors: $d(N) = (a+1)(b+1)(c+1)$
Sum of factors: $σ (N) = \frac{^{}}{}$

4. Factorials

$N! = N(N-1)(N-2)\cdots 2\cdot 1$ , with $0! = 1! = 1$ .

Trailing zeros in $N!$ : $Z = \left\lfloor\frac{N}{5}\right\rfloor + \left\lfloor\frac{N}{25}\right\rfloor + \left\lfloor\frac{N}{125}\right\rfloor + \cdots$

5. Remainder & Divisibility Theorems

$a^{n} - b^{n}$ is always divisible by $a - b$ .
$a^{n} + b^{n}$ is divisible by .

6. Unit Digit (Cyclicity)

Only unit digits matter in a product — e.g. $37\times 48\times 29 \Rightarrow 7\times 8\times 9 = 504 \Rightarrow$ unit digit 4.

Last digit	Cycle	Length
0, 1, 5, 6	itself	1
4	4, 6	2
9	9, 1	2
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4

Use $n \bmod 2$ for length-2 digits and $n \bmod 4$ for length-4 (if the remainder is 0, use the last value in the cycle).

7. Number of Digits

Digits in $N$ : $\lfloor \log_{10} N \rfloor + 1$ . Digits in $N^{k}$ : .

8. Base Conversion

Decimal → base $b$ : repeated division by $b$ (read remainders bottom-up).
Base $b$ → decimal: $(a_n a_{n-1}\dots a_0)_b = a_n b^{n} + a_{n-1} b^{\,n-1} + \cdots + a_0$ .

9. Handy Properties

$\gcd(n, n+1) = 1$ · $\gcd(2n, 2n+2) = 2$

p

a + 1

-

1

p - 1

\cdot

\frac{q^{b + 1} - 1}{q - 1}

\cdot

\frac{r^{c + 1} - 1}{r - 1}

\sigma(N) = \dfrac{p^{a+1}-1}{p-1}\cdot\dfrac{q^{b+1}-1}{q-1}\cdot\dfrac{r^{c+1}-1}{r-1}