Episode 8 — Aptitude and Reasoning / 8.6 — Number System

8.6 Number System -- Quick Revision Sheet

Use this sheet for last-minute revision before exams. Everything is condensed into tables, diagrams, and formulas.

1. Number Classification at a Glance

REAL NUMBERS
├── RATIONAL NUMBERS (can be written as p/q, q != 0)
│   ├── INTEGERS (..., -2, -1, 0, 1, 2, ...)
│   │   ├── WHOLE NUMBERS (0, 1, 2, 3, ...)
│   │   │   └── NATURAL NUMBERS (1, 2, 3, 4, ...)
│   │   └── Negative integers (-1, -2, -3, ...)
│   └── Non-integer rationals (1/2, -3/4, 0.75, 0.333...)
└── IRRATIONAL NUMBERS (sqrt(2), pi, e, non-repeating decimals)

Set	Starts from	Includes zero?	Includes negatives?	Example
Natural (N)	1	No	No	1, 2, 3
Whole (W)	0	Yes	No	0, 1, 2
Integer (Z)	-infinity	Yes	Yes	-3, 0, 5
Rational (Q)	--	Yes	Yes	1/2, -7, 0.6
Real (R)	--	Yes	Yes	sqrt(2), pi, -3

2. Prime Numbers up to 100

 2   3   5   7  11  13  17  19  23  29
31  37  41  43  47  53  59  61  67  71
73  79  83  89  97

Total: 25 primes

Key facts:

2 is the only even prime.
1 is neither prime nor composite.
Every prime > 3 is of the form 6k+1 or 6k-1.
To test if N is prime, check divisibility by primes up to sqrt(N).

3. Even/Odd Rules

Operation	Even + Even	Odd + Odd	Even + Odd
Add/Sub	Even	Even	Odd

Operation	Even x Even	Odd x Odd	Even x Odd
Multiply	Even	Odd	Even

Key: Product is odd ONLY if ALL factors are odd. One even factor makes the product even.

4. Divisibility Rules Table

Divisor	Rule	Example: 5765832
2	Last digit even	2 is even --> Yes
3	Digit sum divisible by 3	5+7+6+5+8+3+2=36, 36/3=12 --> Yes
4	Last 2 digits divisible by 4	32/4=8 --> Yes
5	Last digit 0 or 5	2, neither --> No
6	Divisible by 2 AND 3	Yes and Yes --> Yes
7	Double last digit, subtract from rest	Complex check
8	Last 3 digits divisible by 8	832/8=104 --> Yes
9	Digit sum divisible by 9	36/9=4 --> Yes
10	Last digit 0	No
11		Odd-place sum - Even-place sum

Composite divisibility (use co-prime pairs):

12 = 4 and 3        15 = 3 and 5        18 = 2 and 9
24 = 8 and 3        36 = 4 and 9        72 = 8 and 9

5. Factor Formulas

Given N = p1^a x p2^b x p3^c x ...:

What	Formula
Number of factors	(a+1)(b+1)(c+1)...
Sum of factors	[(p1^(a+1)-1)/(p1-1)] x [(p2^(b+1)-1)/(p2-1)] x ...
Product of all factors	N^(d/2) where d = number of factors
Number of odd factors	(b+1)(c+1)... [ignore power of 2]
Number of even factors	Total - Odd = a(b+1)(c+1)...
Perfect square factors	(floor(a/2)+1)(floor(b/2)+1)(floor(c/2)+1)...
Perfect cube factors	(floor(a/3)+1)(floor(b/3)+1)(floor(c/3)+1)...
Pairs as product of 2	d/2 (if not perfect square) or (d+1)/2 (if perfect square)

Quick example:

360 = 2^3 x 3^2 x 5^1

Total factors   = 4 x 3 x 2 = 24
Odd factors     = 3 x 2 = 6
Even factors    = 24 - 6 = 18
Sq. factors     = 2 x 2 x 1 = 4
Factor pairs    = 24/2 = 12

6. Unit Digit Cycles

+-------+-------------------+---------+
| Digit | Cycle             | Length  |
+-------+-------------------+---------+
|   0   | 0                 |   1     |
|   1   | 1                 |   1     |
|   2   | 2, 4, 8, 6        |   4     |
|   3   | 3, 9, 7, 1        |   4     |
|   4   | 4, 6              |   2     |
|   5   | 5                 |   1     |
|   6   | 6                 |   1     |
|   7   | 7, 9, 3, 1        |   4     |
|   8   | 8, 4, 2, 6        |   4     |
|   9   | 9, 1              |   2     |
+-------+-------------------+---------+

How to use: Divide the exponent by cycle length. If remainder = r, take position r in cycle. If remainder = 0, take the last position.

Example: 8^75
Cycle of 8: {8, 4, 2, 6}, length 4
75 mod 4 = 3
3rd position = 2
Unit digit = 2

7. Remainder Shortcuts

Rem(a + b, n) = [Rem(a, n) + Rem(b, n)] mod n
Rem(a x b, n) = [Rem(a, n) x Rem(b, n)] mod n
Rem(a^k,   n) = [Rem(a, n)]^k mod n

Power patterns:

(n-1)^even mod n = 1
(n-1)^odd  mod n = n - 1
(n+1)^any  mod n = 1

Theorems:

Theorem	Condition	Result
Fermat's Little	p prime, gcd(a,p)=1	a^(p-1) mod p = 1
Euler's	gcd(a,n)=1	a^phi(n) mod n = 1
Wilson's	p prime	(p-1)! mod p = p-1

8. Euler's Totient (phi) Quick Values

phi(1)  = 1      phi(2)  = 1      phi(3)  = 2
phi(4)  = 2      phi(5)  = 4      phi(6)  = 2
phi(7)  = 6      phi(8)  = 4      phi(9)  = 6
phi(10) = 4      phi(11) = 10     phi(12) = 4
phi(15) = 8      phi(20) = 8      phi(24) = 8
phi(30) = 8      phi(36) = 12     phi(100)= 40

Formula: phi(N) = N x (1 - 1/p1) x (1 - 1/p2) x ...

9. Trailing Zeros in n!

Zeros = floor(n/5) + floor(n/25) + floor(n/125) + floor(n/625) + ...

n	Trailing Zeros
10!	2
20!	4
25!	6
50!	12
100!	24
200!	49
250!	62
500!	124
1000!	249

10. Highest Power of Prime p in n!

Power = floor(n/p) + floor(n/p^2) + floor(n/p^3) + ...

For composite d = p^a x q^b x ..., find the power of each prime in n!, divide by the required exponent, and take the minimum.

11. Sum Formulas

Sum of first n naturals      = n(n+1)/2
Sum of first n even numbers  = n(n+1)
Sum of first n odd numbers   = n^2
Sum of squares (1 to n)      = n(n+1)(2n+1)/6
Sum of cubes (1 to n)        = [n(n+1)/2]^2

12. Perfect Square Checks

Can end in: 0, 1, 4, 5, 6, 9
Cannot end in: 2, 3, 7, 8

mod 4: remainder is 0 or 1 (never 2 or 3)
mod 3: remainder is 0 or 1 (never 2)

If ends in 0: even number of trailing zeros
If ends in 5: must end in 25
All exponents in prime factorization are even
Always has ODD number of factors

13. Useful Algebraic Identities

a^n - b^n is always divisible by (a - b)
a^n + b^n is divisible by (a + b) when n is ODD
a^n - 1 is divisible by (a - 1) for all n

14. Key Numbers to Remember

1001 = 7 x 11 x 13    (abcabc = abc x 1001)
111  = 3 x 37
1111 = 11 x 101
10001 = 73 x 137

Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Powers of 3: 1, 3, 9, 27, 81, 243, 729
Powers of 5: 1, 5, 25, 125, 625, 3125

15. Counting in Ranges

Integers from a to b inclusive     = b - a + 1
Multiples of k in [1, n]          = floor(n/k)
Multiples of k in [a, b]          = floor(b/k) - floor((a-1)/k)
Neither div by m nor n in [1, N]  = N - floor(N/m) - floor(N/n) + floor(N/lcm(m,n))

16. Common Exam Patterns -- One-Line Answers

Pattern	Formula
Largest n-digit multiple of k	k x floor((10^n - 1)/k)
Smallest n-digit multiple of k	k x ceil(10^(n-1)/k)
Remainder of n! / p (p prime, p <= n)	0
Remainder of (1!+2!+...+n!) / k	Only sum up to (k-1)!
Product of k consecutive integers	Always divisible by k!
abcabc	Always divisible by 7, 11, 13
Last two digits of a^n (gcd(a,100)=1)	Reduce power mod 40

17. Quick Revision Checklist

Before the exam, make sure you can: