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Important Formulas of Number System


Formulas of Number Series
  1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
  2. (12 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
  3. (13 + 23 + 33 + ..... + n3) = (n(n + 1)/ 2)2
  4. Sum of first n odd numbers = n2
  5. Sum of first n even numbers = n (n + 1)

Mathematical Formulas
  1. (a + b)(a - b) = (a2 - b2)
  2. (a + b)2 = (a2 + b2 + 2ab)
  3. (a - b)2 = (a2 + b2 - 2ab)
  4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
  5. (a3 + b3) = (a + b)(a2 - ab + b2)
  6. (a3 - b3) = (a - b)(a2 + ab + b2)
  7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
  8. When a + b + c = 0, then a3 + b3 + c3 = 3abc
  9. (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn

Shortcuts for number divisibility check
  1. A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.
  2. A number is divisible by 3, if the sum of its digits is divisible by 3.
  3. A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
  4. A number is divisible by 5, if its unit's digit is either 0 or 5.
  5. A number is divisible by 6, if it is divisible by both 2 and 3.
  6. A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
  7. A number is divisible by 9, if the sum of its digits is divisible by 9.
  8. A number is divisible by 10, if it ends with 0.
  9. A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.
  10. A number is divisible by 12, if it is divisible by both 4 and 3.
  11. A number is divisible by 14, if it is divisible by 2 as well as 7.
  12. Two numbers are said to be co-primes if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x.

Shortcuts for 'recurring decimal to fraction' conversion
  1. For recurring decimals of format '0.abababab...' (ab repeats), equivalent fraction will be "repeating group (here ab)"/"as many 9's as the number of digits in repeating group"
  2. For recurring decimals of format '0.abbbbb...' (b repeats), equivalent fraction will be (entire decimal group - non-repeating decimal group)/(as many 9's as the number of repeating digits in the decimal part with as many 0's as the number of non-repeating digits in the decimal part)