CSAT
Number System
CSAT
Number System — Quick Revision
One-page CSAT revision of the number system: divisibility rules, factor counting, HCF–LCM uses, remainder and unit-digit cycles, trailing zeros, lakh–crore conversions.
SyllabusCSAT · Basic numeracy (numbers and their relations, orders of magnitude) — Class X level
Every formula you need for CSAT number-system questions, on one page.
Divisibility Rules
| By | Test |
|---|---|
| 2 / 5 / 10 | Last digit even / 0 or 5 / 0 |
| 3 / 9 | Digit sum divisible by 3 / 9 |
| 4 / 8 | Last two / last three digits divisible by 4 / 8 |
| 6 / 12 | By 2 and 3 / by 3 and 4 |
| 7 | Double last digit, subtract from the rest, repeat |
| 11 | Alternating digit sum is 0 or a multiple of 11 |
Factors, HCF and LCM
| Rule | Formula |
|---|---|
| Number of factors of $p^a q^b r^c$ | $(a+1)(b+1)(c+1)$ |
| HCF / LCM | Lowest / highest powers of primes |
| Two numbers | HCF × LCM = product |
| Recur together | LCM |
| Smallest number leaving remainder $r$ | LCM $+ r$ |
| Largest number leaving remainder $r$ in each | HCF of (each number $- r$) |
| Equal pieces, largest size | HCF |
Remainders and Unit Digits
| Rule | Value |
|---|---|
| Remainder of product | Product of remainders, reduced again |
| Divisor of divisor | Remainder by $D$ is $r$, $d$ divides $D$ → remainder by $d$ = $r \bmod d$ |
| Unit-digit cycles of 2, 3, 7, 8 | Length 4: (2,4,8,6), (3,9,7,1), (7,9,3,1), (8,4,2,6) |
| Unit-digit cycles of 4, 9 | (4,6), (9,1) |
| 0, 1, 5, 6 | Unchanged |
| Exponent multiple of 4 | Use the 4th term |
| Trailing zeros of $n!$ | $\lfloor n/5 \rfloor + \lfloor n/25 \rfloor + \lfloor n/125 \rfloor + \cdots$ |
High-yield facts
- 1 is neither prime nor composite; 2 is the only even prime; 25 primes below 100.
- Product of three consecutive integers is divisible by 6.
- Square of an odd number leaves remainder 1 on division by 8.
- Primes above 3 are of the form $6k \pm 1$.
- $1 + 2 + \cdots + n = \dfrac{n(n+1)}{2}$; sum of the first $n$ odd numbers $= n^2$.
Orders of Magnitude
| Indian | Power of 10 | International |
|---|---|---|
| 1 lakh | $10^5$ | 100 thousand |
| 10 lakh | $10^6$ | 1 million |
| 1 crore | $10^7$ | 10 million |
| 100 crore | $10^9$ | 1 billion |
| 1 lakh crore | $10^{12}$ | 1 trillion |
Traps
Three classic errors
- Using HCF × LCM = product for three numbers.
- Taking the “zeroth” term when the exponent is a multiple of 4.
- Forgetting the extra fives from 25, 125… in trailing-zero questions.
Related Topics
- Number System: Divisibility, HCF–LCM and Remainders
- Number System — practice questions
- Percentages, Ratio and Averages — quick revision
- All quick-revision sheets
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