2. Powers of 2 2 Power of 2 –Any number from 1 to M can be can be written as the sum of one or more of the integers and every integer being used only once. The Set of integers used in this case would be the numbers formed using ALL powers of 2 up to the power of 2 which gives the number less than M. For Example – Any number from 1 to Number 9 – It can be formed as sum of 1+2+4+8 i.e. power of 2 being from 1 to 3. 1 = 20 2 = 21 3 = 20 + 21 4 = 22 5 = 20 + 22 6 = 21 + 22 7 = 20 + 21 + 22
3. Powers of 3 3 Power of 3 –Any number up to 40 can be formed using powers of 3, and using either addition or subtraction. For Example – 14 = - 1 – 3 - 9 + 27 = - 30 – 31 - 32 + 33
4. Multiples 4 Any Multiple of 2 will end with (have at units place) either 0,2,4,6,8 For any multiple of 4, the last 2 digits are divisible by 4. For example – 424 is divisible by 4; 112 is divisible by 4 and so on. For any Multiple of 5, the last digit is either 0, 0r 5. Any multiple of 10 will have last digit as 0. Any multiple of 6 will end with 0,2,4,6,8 AND will be divisible by 3.
5. Multiples 5 For multiplying by 5 (51), multiply the number by 10 and divide by 2 (21). For multiplying by 25 (52), multiply the number by 100 and divide by 4 (22). For multiplying by 125(53), multiply the number by 1000 and divide by 8 (23). For multiplying by 11, multiply by 10 and add the number itself to the result. For Example – Multiply 121 by 11 = 121*10 = 1210 + 121 = 1331. For multiplying by 12 treat it as multiplying by 10 and add the double of the number to the result.
6. Multiples 6 Similarly to find a product with a number closer to the higher multiple of 10, say 16, 18 etc. Multiply by 20 and then adjust for the product derived at by multiplying with the unit difference. Example – multiply 324 by 18 = 324 ( 20-2) = 6480 – 648 = 5832. Hence, FOR EASE, TRY TO MULTIPLY BY A ROUND NUMBER AND (20) THEN MULTIPLY BY THE BALANCE NUMBER (2) ADD or SUBTRACT the Products (6480 – 648).
7. Squares 7 Any square of a number ending in 1 will end in 1 and at the tens place will have the double of the original number at the tens place. Example = 123451*123451 = 15240149401 51*51 = 2601 41*41 = 1681 Any number ending with 1,2,3,4,5,6,7,8,9,0 will have a square that ends in the same number as would the square of 1,2,3,4,5,6,7,8,9,0 would end. Example – Square of 78237492 would end in 4, square of 89239297 would end in 9 and so on.
8. Others 8 The nth Odd number = (2n-1). Example- 3rd odd number is 5 which is 2*3 – 1 = 6-1. Also, for any number n, n2 = sum of the first n odd numbers. Example = 42 = sum of 1st 4 odd numbers = 1+3+5+7 = 16
9. Fractions 9 For any fractions, Let N = numerator and D = Denominator If N is same for all Smaller the D, higher the Fraction Inverse Relation If D is same for all higher the N, higher the Fraction Direct relation If D-N > 0 and is same for all Fraction with higher values of N or D will be higher If D-N < 0 and is same for all Fraction with lower values of N or D will be higher Fraction with highest N and smallest D will be the highest and vice versa.
