Numbers - Important Formulae

Important Formulae

1. Prime Numbers - A number which doesn't have any factor apart from one and itself. Ex: 2,3,5, etc.

• Every prime number greater than 3 can be written in the form of (6k + 1) or (6k - 1) where k is an integer.
• To find a number N as prime number or not, let k be the smallest number such that k² ≥ N. Now, test whether N is divisible by any of the prime numbers less than or equal to k.

    Composite Numbers - Any number other than 1, which is not a prime number. Ex: 4,6,8, etc.
    Relative Primes - Two numbers are said to be relative primes or co-primes if they do not have any common factor other than 1. For Ex: 15 and 16 are relative prices.
    Perfect Numbers - If the sum of ALL its factors excluding itself (but including 1) is equal to the number itself. For Ex: 6,28,496, etc.

2. Even and Odd Numbers - Numbers divisible by 2 are even numbers and not divisible by 2 are odd numbers.

• Sum of any number of even numbers is always even.
• The sum of odd number of odd numbers (i.e.sum of 3 odd numbers) is always odd whereas the sum of even number of odd numbers (i.e. sum of 2 odd numbers) is always even.
• The product of any number of odd numbers is always odd.
• The product of any number of numbers where there is atleast one even number is even.

3. Order of Arithmetic Operations- BODMAS = Brackets Of Division Multiplication Addition Subtraction
    Order of opening brackets - simple brackets ( ), curly brackets { }, square brackets [ ]

4. Rules for Divisibility
    Divisibility by 2 - even number as its last digit
    Divisibility by 3 - sum of its digits is a multiple of 3
    Divisibility by 4 - number formed with its last two digit is divisible by 4
    Divisibility by 5 - last digit is 5 or zero
    Divisibility by 6 - divisible both by 2 and 3
    Divisibility by 8 - last 3 digits of the number is divisible by 8
    Divisibility by 9 - sum of its digits is a multiple of 9
    Divisibility by 10 - last digit is zero
    Divisibility by 11 - difference between sum of digits in odd places and sum of digits in even places should be zero or a multiple of 11
    Divisibility by 12,14,15 - taking factors of the number which are relatively prime and checking the divisibility of the number by each of the factors. For Ex: a number is divisible by 14 if it is divisible both by 2 and 7.

5. Recurring Decimals
    0.454545454... = 45 is recurring. This can be represented as fraction = 45 / 99
    (recurring digits) / (number of 9s as the number of recurring)
    0.1474747474747...1 is constant and only 47 is recurring. This can be represented as fraction =
    (147 - 1)/990
    (number - non_Recurring_part) / ( number of 9s as the number of recurring, appended with number of zeros as number of digits that are not recurring)

6. Important algebraic formulae:
    (a + b)²        = a² + 2ab + b²
    (a - b)²         = a² - 2ab + b²
    (a + b + c)²  = a² +  b² + c² + 2ab + 2bc + 2ca
    (a + b)³        = a³ + b³ + 3ab(a + b)
    (a - b)³         = a³ - b³ - 3ab(a - b)
    a² - b²           = (a + b)(a - b)
    a³ + b³          = (a + b)(a² - ab + b²)
    a³ - b³           = (a - b)(a² + ab + b²)
    a³ + b³ + c³ - 3abc =  (a + b + c)(a² +  b² + c² - ab - bc - ca)
    a³ + b³ + c³ = 3abc, if a + b + c = 0

7. Few points to remember during Division:
    dividend = the number which is being divided
    divisor = the number by which we divide the dividend
    a. A number when divided by d leaving a remainder of r is of the form dk + r where k is some integer from 0, 1, 2, ....
    b. When a number N is divided by divisor d has a remainder r, then the number N - r is exactly divisible by d.
    c. When a number N is divided by a divisor d has a remainder r, then
      i. N - r will be the largest multiple of d which is less than or equal to N.
      ii. N + (d - r) will be the smallest multiple of d which is greater than N.
    d. When a division is split into a sum/difference of two divisions (with same divisor as original), the original remainder will be equal to sum/difference of the two individual divisions.
    e. If the remainder in a division is negative, then add the divisor to the negative remainder to get the positive remainder.
    f. In a division, if the dividend is multiplied by a certain factor and then divided by the same divisor, then the new remainder will be obtained by multiplying the original remainder by same factor with which the dividend has been multiplied.
    g. If the remainder is greater than the divisor, it means division is not complete.

8. LCM and HCF
    LCM = least number which is divisible by each of the numbers
    HCF or GCD = largest factor of the numbers

• Product of two numbers = LCM x HCF
• HCM of fractions = HCF of numerators / LCM of denominators
• LCM of fractions = LCM of numerators / HCF of denominators
• Any number which when divided by p, q or r leaving the same remainder s in each case will be of the form k (LCM of p, q and r) + s where k = 0, 1, 2, ... If we take k=0, then we get the smallest such number.
• Any number which when divided by p, q or r leaving respective remainders of s, t and u where (p-s) = (q-t) = (r-u) = v (say), will be of the form k (LCM of p, q and r) - v. The smallest such number will be obtained by substituting k = 1.
• The largest number with which the numbers p, q or r are divided giving remainders of s, t and u respectively will be the HCF of the three numbers of the form (p-s), (q-t) and (r-u)
• The largest number with which if we divide the numbers p, q and r, the remainders are the same will be the HCF of (p-q) and (p-r)

9. Last Digit of Any Power:
    The last digits of the powers of any number follow a cyclic pattern - they repeat after certain number of steps. If we find out after how many steps the last digit of the powers of a number repeat, then we can find out the last digit of any power of any number.

10. Last Digit of a Sum or Product:
      We can just take the last digit of the numbers and add/product them.

11. Remainder in Divisions involving powers of numbers
     The remainders of powers of a number also follow a certain pattern. If we identify the pattern in which the remainders repeat, we can find out the remainder of any division given.

12. Rules pertaining to aⁿ + bⁿ or aⁿ - bⁿ
      aⁿ - bⁿ

• It is always divisible by a - b.
• When n is even it is also divisible by a + b.
• When n is odd it is not divisible by a + b.

     aⁿ + bⁿ

• It is never divisible by a - b
• When n is odd it is divisible by a + b.
• When n is even it is not divisible by a + b.

13. Useful points in solving number problems

• When any two consecutive integers are taken one of them is odd and the other is even. Hence, the product of any two consecutive integers is always even i.e. divisible by 2.
• Out of any 3 consecutive integers, one of them is divisible by 3 (Atleast one of the three is definitely even.). Hence, the product of any 3 consecutive integers is always divisible by 6.
• Out of any n consecutive integers, exactly one number will be divided by n and the product of n consecutive integers will be divisible by n!