I. Squares and square roots: From 12to 302.

Squares: 1 - 50
12 = = 676
1 262
22 = = 729
4 272
32 = = 784
9 282
42 = = 841
16 292
52 = = 900
25 302
62 = = 961
36 312
72 = = 1024
49 322
82 = = 1089
64 332
92 = = 1156
81 342
102 = = 1225
100 352
112 = = 1296
121 362
122 = = 1369
144 372
132 = = 1444
169 382
142 = = 1521
196 392
152 = = 1600
225 402
162 = = 1681
256 412
172 = = 1764
289 422
182 = = 1849
324 432
192 = = 1936
361 442
202 = = 2025
400 452
212 = = 2116
441 462
222 = = 2209
484 472
232 = = 2304
529 482
242 = = 2401
576 492
252 = = 2500
625 502
II. Cubes and cubic roots: From 13to 123.

1 - 15

13 = 1 63 = 216 113 =
1331
23 = 8 73 = 343 123 =
1728
33 = 27 83 = 512 133 =
2197
43 = 64 93 = 729 143 =
2744
53 = 125 103 = 1000 153 =
3375
III. Powers of 2: From 21to 212.
2 1
= 2 31 = 3
22 = 4 32 = 9
23 = 8 33 = 27
24 = 16 34 = 81
25 = 32 35 = 243
26 = 64 36 = 729
 27 = 128 37 = 2187
28 = 256 3 8
= 6561
29 = 512 39 = 19683
210 = 1024 310 = 59049
211 = 2048 311 = 177147
212 = 4096 312 = 531441
IV. Prime numbers from 2 to 127: It also helps to know the primes in the 100's, like
113, 127, 131, ... It's important to know not just the primes, but why 51, 87, 91,
and others are not primes (especially 1).

2 97 227 367
3 101 229 373
5 103 233 379
7 107 239 383
11 109 241 389
13 113 251 397
17 127 257 401
19 131 263 409
23 137 269 419
29 139 271 421
31 149 277 431
37 151 281 433
41 157 283 439
43 163 293 443
47 167 307 449
53 173 311 457
59 179 313 461
61 181 317 463
67 191 331 467
71 193 337 479
73 197 347 487
79 199 349 491
83 211 353 499
89 223 359

Dividing by 11

Let's look at 352, which is divisible by 11; the answer is 32. 3+2 is 5;
Another way to say this is that 35 minus 2 is 33.

Now look at 3531, which is also divisible by 11. It is not a coincidence.

that 353-1 is 352 and 11 × 321 is 3531.

Dividing by 7
 To find out if a number is divisible by seven, take the last digit, double it,
and subtract it from the rest of the number.
If you had 203, you would double the last digit to get six, and
subtract that from 20 to get 14. If you get an answer divisible by 7
including zero), then the original number is divisible by seven. If you
If you don't know the new number's divisibility, you can apply the rule again.

Dividing by 12

Check for divisibility by 3 and 4.

Dividing by 13

Here's a straightforward method supplied by Scott Fellows:

Delete the last digit from the given number. Then subtract nine times the
deleted digit from the remaining number. If what is left is divisible by 13,
then so is the original number.

For any prime p (except 2 and 5), a rule of divisibility could be 'created'
using this method:

1. Find m, such that m is the (preferably) smallest multiple of p that ends in either 1
or 9.
Delete the last digit and subtract (if multiple ends in 9) or
add (if it ends in 1) the deleted digit times the integer
nearest to m/10. For example, if m = 91, the integer closest
to 91/10 = 9.1 is 9; and for 3.9, it's 4.
3. Verify if the result is a multiple of p. Use this process until
It's obvious.

Example 1: Let's see if 14281581 is a multiple of 17.

In this case, m = 51 (which is 17*3), so we'll be deleting the last number and
subtracting it fivefold.

1428158 - 5*1 = 1428153

142815 - 5*3 = 142800
14280 - 5*0 = 14280
1428 - 5*0 = 1428
142 - 5*8 = 102
10 - 5*2 = 0, which is a multiple of 17, so 14281581 is multiple of 17.
 Example 2: Let's see if 7183186 is a multiple of 46.
First, note that 46 is not a prime number, and its factorization is 2*23. So,
7183186 needs to be divisible by both 2 and 23. Since it's an even number, it's
obviously divisible by 2.
So let's verify that it is a multiple of 23:

m = 3*23 = 69, which means we'll be adding the deleted digit sevenfold.
718318 + 7*6 = 718360
71836 + 7*0 = 71836
7183 + 7*6 = 7225
722 + 7*5 = 757
75 + 7*7 = 124
12 + 7*4 = 40
4 + 7*0 = 4 (not divisible by 23), so 7183186 is not divisible by 46.

Note that you could've stopped calculating whenever you find the result to be
obvious (i.e., you don't need to do it until the end). For example, in example 1 if
you recognize 102 as divisible by 7, you don't need to continue (likewise, if you
recognized 40 as not divisible by 23.
The idea behind this method is that you're either subtracting m*(last digit) and
then dividing by 10, or adding m*(last digit) and then dividing by 10.

V. Sum of the numbers in an arithmetic series: In an arithmetic series the

The difference between terms is a constant. Example: 4 + 10 + 16 + 22 + ... + 100 is
an arithmetic series. The formula for the sum is

(a + z) / 2*

where n is the number of terms in the sequence, a is the lowest term, and z is the
highest term. Finding the sum of the above sequence:

17 (4 + 100) / 2 = l7 * 104 / 2 = 884

Why is it equal to 17? Figure it out.

x(n - 1) + a = z where x is the amount being added to each number, n is the

number of terms, a is the lowest term, and z is the highest term. The minus 1 is to
account for the first term. Using the sequence above this would look like this:

6(n - 1) + 4 = 100, subtract 4 from both sides, so 6(n - 1) = 96, divide both sides.
by 6, so(n - 1)= 16, add 1 to both sides so,n= 17

If you recognize this formula, it is also known as Gauss' formula. The story behind
It is actually very interesting, while Gauss was in school (it was a one room
 school house) his teacher gave them busy work; they had to add all of the
Numbers between 1 and 100 on a chalkboard, it took Gauss only a few seconds.
because he found the pattern that forms when you add the first and last, second
and second to last numbers and so on you get the same number every time so you
only have to take half of this number (which turns out to be the average of all)
of the numbers in the sequence) and multiply that by the number of numbers.
Since you are multiplying the average by the number of numbers, you get the
answer.

VI. Triangle or triangular numbers: 3, 6, 10, 15, 21, 28, 36, 45, 55 are triangle
numbers. (This is based on V. above!)

...

n (n + 1) / 2
(it can be used to find the number of dots)

To find 1 + 2 + 3 + 4 + 5 + ... + n, just multiply it by its next higher consecutive.

number, and divide by 2.

Example: find 1+2+3+4+5+...+28.

28 (28 + 1) / 2 = 406

VII. Pythagorean Theorem: Applications of this famous relationship occur very

often in math competition and on the S.A.T.

a2+ b2= c2

A.Pythagorean Triples: Integral values of a, b, and c, where a, b, and c are

relatively prime (one or more of the numbers is/are prime so it can't be)
reduced in size)
3 - 4 - 5 (the most common)
5 - 12 - 13 15 - 112 - 113
7 - 24 - 25 16 - 63 - 65
8 - 15 - 17 17 - 144 - 145
9 - 40 - 41 19 - 180 - 181
11 - 60 - 61 20 - 21 - 29
12 - 35 - 37 20 - 99 - 101
13 - 84 - 85
B.The 45o- 45o90oright triangle, or right isosceles triangle: This is half
of a square, where the legs are congruent. If the leg is s, then the
The hypotenuse is * sqrt(2) (times the square root of 2). Example: A square
has a perimeter of 10, and you need to know the length of the diagonal.
 s = 2.5, so d = 2.5 * sqrt(2).

C.The 30o- 60o- 90oright triangle: This is half of an equilateral triangle.

The short leg, the one opposite the 30oangle, iss, the hypotenuse is 2s,
and the long leg, which is opposite the 60oangle, is * sqrt(3)(stimes the
square root of 3)
II. Number of diagonals in an s-sided polygon: I've seen so many different
applications of this formula:

(s (s - 3)) / 2

where s is the number of sides of the polygon. A polygon having 45 sides has
45*(45-3)/2 = 945 diagonals.

III. Fraction, decimal, percent equivalencies: You must know these backward.
forward, and upside down. The halves, thirds, fourths, fifths, sixths, sevenths,
(yes, sevenths!), eighths, ninths, tenths, (so hard, he?), elevenths, twentieths
twenty-fifths, fiftieths. It also helps to know the twelfths, fifteenths, and sixteenths.
You should, for example, be able to recognize, instantly and without hesitation,
that 83 1/3% is 5/6, and that 9/11 is 81 9/11%.
IV. Diagonal of a cube:

s * sqrt(3)

Where is the edge of the cube? This is an application of the Pythagorean.

Theorem: See Section VII(B) above. Figure out why this is so. Don't expect me to
do it for you.

V.Area and Volume:

A. Area of a square, given the side:A = s2
B. Area of a square, given diagonal:A = d2/2
C. Area of a rhombus, given diagonals:A = (d1d2)/2
(B and C are closely related. How?)
D. Area of triangle:A = (bh) / 2
E. Area of circle:A = π r2
F. Area of trapezoid:A = 1/2 h (b1+ b2)
G. Volume of cylinder and prism:V = B h
H. Volume of cone and pyramid:V = 1/3 (B h)
I. Volume of a sphere:V = 4/3 π r3
J.Surface area of a sphere:A = 4 π r2
I also expect you to know the following procedures:
A.Scientific notation, both multiplying and dividing numbers written in this
form. All you do is apply the rules you've learned about exponents.
B.Turning arepeating decimal into a simple fraction. You see this almost
every week; isn't it time to learn the shortcut for this, once and for all?
 C.Turning a20 5/6% = 123/600 or 41/200
5/24
D.Setting upprobability problems. This is usually plain, simple reading.
Know the terms 'with replacement'; 'without replacement', 'at least one'.
E. Be able to generatePascal's Triangle on the spot. There are so many
applications of this in combinations and probability.
F. Can you think of any more? I can. You should.

This is just the beginning. If you think you can't memorize the relationships and formulas
in this "Bible", you are absolutely right. Chances are the person ranked above you knows
it better than you do. If, on the other hand, you think this can be done, you're quite right!