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) |