Division and divisibility

If dividend = O, Divisor = P, Quotient = Q and remainder = R,

O = P x Q + R.

 

Hence, we can say that O mod Q = R. In other words, Q modulus of O is R.

Ex: 6 mod 3 = 0; 7 mod 3 = 1; 8 mod 3 = 2; 9mod 3 =0; 24 mod 11 = 2.

To find divisibility by certain integers here are a few shortcuts:

Divisible by 2: The dividend should be even. It should end with 0, 2, 4, 6, and 8.

Divisible by 4: The last two digits of the dividend is divisible by 4, i.e., 7028, 494464, 759988, 54672, etc.

Divisible by 8: Last 3 digits are divisible by 8. For example, 432000, 12323640, 214016, etc.

Divisible by 2^n: Last n digits are divisible by 2^n. ( Generalization from above concepts ).

Divisible by 3: Sum of digits of the dividend is divisible by 3. For example, in 45816732, 4+5+8+1+6+7+3+2 = 36. 36 mod 3 = 0, i.e., 36 is divisible by 3, hence 45816732 is divisible by 3.

Divisible by 5: The last digit is divisible by 5, i.e., the last digit is 0 or 5.

Divisible by 7: Say the number is 32116. 3211-6*2=3199. 319-9*2=301. 30-2*1=28, this is 7*4. This processing a number can be stopped at any step where we can be sure if the resultant value is divisible by 7 or not.

Divisible by 9: The sum of digits is divisible by 9. For example, 3927537. 3+9+2+7+5+3+7=36 this is divisible by 9. Hence, 3927537 is divisible by 9.

Divisible by 10^n: The last digit of the number has n zeroes.

Divisible by 11: Difference between the sum of digits at odd places and sum of digits at even places is equal to 0 or is divisible by 11. For example, in 1078, 1+7=0+8. Hence, 1078 is divisible by 11.

Divisible by 13: Say we have a number 87724. 8772+4*4=8788; 878+8*4=910; 91+4*0=91; 9+1*4=13. Hence, 87724 is divisible by 13.

Divisible by 17: Say we have a number 12716. 1271-6*5=1241. 124-1*5=119. 11-9*5=-34. Hence 12716 is divisible by 17.

Shortcut for divisibility by 7, 11, 13:
If the number is n=1000*x + y, then n is divisible by 7, 11 and 13 if |x-y| is divisible by 7, 11 and 13 respectively.