Multiplication Table 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Square Roots 2 = 1.41 3 = 1.73 5 = 2.236 6 = 2.449 7 = 2.646 10 =3.16

Squares and Cubes Number ( x ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25 Square ( x 2 ) 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 441 484 529 576 625 Cube ( x 3 ) 1 8 27 64 125 216 -

Fractions and Percentage Fraction 1/2 1/3 2/3 1/4 3/4 1/5 2/5 3/5 4/5 1/6 5/6 1/8 3/8 5/8 7/8 1/9 2/9 1 / 10 1 / 20 1 / 100 Decimal 0.5 0.33 0.66 0.25 0.75 0.2 0.4 0.6 0.8 0.166 0.833 0.125 0.375 0.625 0.875 0.111 0.222 0.1 0.05 0.01 Percentage 50 33 1/3 66 2/3 25 75 20 40 60 80 16 2/3 83 1/3 12 1/2 37 1/2 62 1/2 87 1/2 11 22 10 5 1

Natural numbers are 1,2,3 Whole numbers are 0,1,2,3 Integers are 0,1, 2, 3, Composite number is opposite of prime number Rational number can be expressed as p/q Irrational number cannot be expressed as p/q such as 2 A fraction will result in a terminating decimal only if the denominator is written in form of prime numbers of 2 and 5. Otherwise the fraction will not result in a terminating decimal.

Prime Numbers 0 and 1 are not prime numbers. Here is a table of all prime numbers up to 1,000: 2 3 5 7 11 13 17 19 23

29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997

Method to determine a prime number: Find approx square root of the number. Then check if all the prime numbers below the square root are factors of the given number. If none are then the number is prime else not. e.g. number 91. approx sq root is 10 Prime number below 10 are 2.3.5&7. 91 is not divisible by2,3 or 5. But it is divisible by 7. Therefore 91 is not prime.

Divisibility Divisibility by 2, 4, 8, 16, 32.. A number is divisible by 2, 4, 8, 16, 32,2 n when the number formed by the last one, two, three, four, five...n digits is divisible by 2, 4, 8, 16, 32,..2 n respectively. i.e 1246384 is divisible by 8 because the # formed by the last 3 digits i.e. 384 is divisible by 8.

Divisibility by 3 and 9 A number is divisible by 3 or 9 when the sum of the digits of the number is divisible by 3 or 9 respectively. i.e. 3144 is divisible by 3 because the sum of the digits- 3 + 1 + 4 + 4 = 12 is divisible by 3. i.e. 8406 is divisible by 9 because the sum of the digit- 8 + 4 + 0 + 6 = 18 is divisible by 9.

Divisibility by 5, 10 Divisible by 5 if the number ends in 0 or 5 Divisible by 10 if the number ends in 0

Divisibility by 11 11 - Start with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11. eg. Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11. When any number with even number of digits is added to its reverse, the sum is always divisible by 11. e.q: 1234+4321 is divisible by 11

Divisibility by 7, 11, and 13 Let a number be ....kjihgfedcba where a, b, c, d, are respectively units digits, tens digits, hundreds digits, thousands digits and so on. Starting from right to left, we make groups of three digit numbers successively and continue till the end. It is not necessary that the leftmost group has three digits. Grouping of the above number in groups of three, from right to left, is done in the following manner kj,ihg,fed,cba We add the alternate groups (1 st , 3 rd , 5 th etc.. and 2 nd , 4 th , 6 th , etc..) to obtain two sets of numbers, N 1 and N 2 . In the above example, N 1 = cba + ihg and N 2 = fed + kj Let D be difference of two numbers, N 1 and N 2 i.e. D = N 1 - N 2 . - If D is divisible by 7, then the original number is divisible by 7. - If D is divisible by 11, then the original number is divisible by 11 - If D is divisible by 13 then the original number is divisible by 13.

Corollary: Any six-digit, or twelve-digit, or eighteen-digit, or any such number with number of digits equal to multiple of 6, is divisible by EACH of 7, 11 and 13 if all of its digits are same . For example 666666, 888888888888 etc. are all divisible by 7, 11, and 13. Example Find if the number 29088276 is divisible by 7. Answer: We make the groups of three as said above- 29,088,276 N 1 = 29 + 276 = 305 and N 2 = 88 D = N 1 - N 2 = 305-88 = 217. We can see that D is divisible by 7. Hence, the original number is divisible by 7.

Divisibility by 25 and 125 A number is divisible by 25 and 125 when the number formed by the last two and three right hand digits are divisible by 25 and 125 respectively. 1025, 3475 divisible by 25 2125, 5375 divisible by 125

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Example: An amount of $1500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years. ( ) =1938.84

When the rates are different for different years A= P(1+R1/100)(1+R2/100)(1+R3/100) Note: No need to calculate the compound interest rate, just calculate the simple interest and the answer will be slightly higher.

Descriptive StatisticsThe average or (arithmetic) mean of n numbers is defined as the sum of the n numbers divided by n. . For example, the average of 6, 4, 7, 10, and 4 is (6+4+7+10)/4 = 6.2 The median is another type of center for a list of numbers. To calculate the median of n numbers, first order the numbers from least to greatest; if n is odd, the median is defined as the middle number, whereas if n is even, the median is defined as the average of the two middle numbers. In the example above, the numbers, in order, are 4, 4, 6, 7, 10, and the median is 6, the middle number. For the numbers 4, 6, 6, 8, 9, 12, the median is (6 + 8)/2. The mode of a list of numbers is the number that occurs most frequently in the list. For example, the mode of 1, 3, 6, 4, 3, 5 is 3. A list of numbers may have more than one mode. For example, the list 1, 2, 3, 3, 3, 5, 7, 10, 10, 10, 20 has two modes, 3 and 10. The degree to which numerical data are spread out or dispersed can be measured in many ways. The simplest measure of dispersion is the range, which is defined as the greatest value in the numerical data minus the least value. For example, the range of 11, 10, 5, 13, 21 is 21 5 = 16. Note how the range depends on only two values in the data.

Standard Deviation1. |Median-Mean| =0. SD is 0 only when the list contains all identical elements (or only 1 element). 6. If we add or subtract a constant to each term in a set: Mean will increase or decrease by the same constant. SD will not change. 7. If we increase or decrease each term in a set by the same percent: Mean will increase or decrease by the same percent.SD will increase or decrease by the same percent. 8. Changing the signs of the element of a set (multiplying by -1) has no effect on SD. 9. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD.

General 1. In order for x and y to be consecutive perfect squares, given that x is greater than y, it would have to be true that: 2. Remember to include '1' if you're asked to count the number of factors a number has 3. Not as tall as means may be shorter or taller. i.e. exactly not the same height as 4. If it is difficult to find solve inequality with modes then square the inequality ... this makes things easier. 5. All squares of even numbers must be multiples of 4 6. If a number is of 2 digit then consider 10x+y kind of equation and so on 7. Sum of integers that are formed by the permutations of n digits is given by equation: = (sum of digits)*(n-1)!*(111... n times) if repetition is not allowed. = (sum of digits)*(n)(n-1)*(111... n times) if repetition is allowed.

i.e What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number? A. 126 B. 1386 C. 3108 D. 308 E. 13986 Here n = 3, sum of digits = 14 thus for 3 digits we have to take 111 only. So sum = 14*3^2*(111)=13986 if repetition is not allowed, then sum = 14 *2!*111 =3108 8. If a+b+c = constant then a2b3c4 have maximum values when a, b and c are in ratio 2:3:4 e.g volume of cylinder V = , and given that r+h=9 then maximum value of V will be when r and h are in ratio 2:1 9. What is remainder when 1421*1423*1425 is divided by 12 classical way to solve this is to multiply all the numbers and then do the division operation to get remainder. however, we can get remainder following way: divide each number with given number separately go get the reminder of each number then multiply, do the operation as many times till resultant number is no more divisible by given number... it's little bit abstract: Find the remainder of 1421*1423*1425 when divided by 12 Rof( 1421*1423*1425) /12 ---------> Rof(5*7*9)/12 = Rof(35*9)/12 Rof(11*9)/12 ---> gives us reminder of 3.

Some math solving strategies: 1. Think for 5 sec before starting calculation 2. Picking numbers is almost always the best to attack even/ odd questions. 3. Generally answers are arranged in ascending order, if picking number is the strategy then checking answer choice C, so it will be clear whether we need to check D&E or A&B 5. First 10 +ve multiple of 5 are: 5,10,15,20,25,30,35,40..........( this is example to show what is multiple of some number means ) 7. e.g. if -2