# Number System Formulas And ShortCut

Important Formulas of Number System

Formulas of Number Series

1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2

(12+ 22+ 32 + ….. + n2) = n ( n + 1 ) (2n + 1) / 6

(13 + 23 + 33 + ….. + n3) = (n(n + 1)/ 2)2

Sum of first n odd numbers = n2

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.

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