Numbers
Basic formulas for Number System
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16) Divisibility by 11: A number is divisible by 11 if the difference between the sum of the digits at odd places and the even places is either 0 or a multiple of 11. Example: 65417, 12211221 For N = 65417 Sum of digits at odd places is 6 + 4 + 7 = 17 Sum of digits at even places is 5 + 1 = 6 Difference is 17 - 6 = 11 Therefore, 65417 is divisible by 11. For N = 12211221 Sum of digits at odd places is 6 Sum of digits at even places is 6 Difference is 6 - 6 = 0 Therefore, 12211221 is divisible by 11. NOTE: If a number is divisible by 'p' as well as by 'q', where 'p' and 'q' are co-primes, then the given number is divisible by 'pq'. Example: 18 is divisible by both 2 and 3. As 2 and 3 are co-primes so 18 is divisible by 6. However, if a number is divisible by 'p' as well as by 'q', where 'p' and 'q' are not co-primes, then the given number may not be divisible by 'pq'. Example: 36 is divisible by both 4 and 6. As 4 and 6 are not co-primes so 36 is not divisible by 24. Some other important formulas1) Sum of first 'n' natural numbers 2) Sum of squares of first 'n' natural numbers  3) Sum of cubes of first 'n' natural numbers  4) N = (Divisor)(Quotient) + (Remainder) |