Key Points - Playing With Numbers

Number in general form: A number is said to be in a general form if it is expressed as the sum of the products of its digits with their respective place values.
Numbers can be written in general form. Thus, a two digit number ab will be written as ab = 10a +b.
The general form of numbers are helpful in solving puzzles or number games.
The reasons for the divisibility of numbers by 10, 5, 2, 9 or 3 can be given when numbers are written in general form.

Tests of Divisiblity: 
(i) Divisibility by 2: A number is divisible by 2 when its one’s digit is 0, 2, 4, 6 or 8.
Explanation: Given number abc = 100a +10b +c. 100a and 10b are divisible by 2 because 100 and 10 are divisible by 2. Thus given number is divisible by 2 only when a = 0, 2, 4, 6 or 8. 

(ii) Divisibility by 3: A number is divisible by 3 when the sum of its digits is divisible by 3. Example: given number = 61785. Sum of digits = 6+1+7+8+5 = 27 which is divisible by 3. Therefore, 61785 is divisible y 3. 

(iii) Divisibility by 4: A number is divisible by 4 when the number formed by its last two digits is divisible by 4. Example: 6216, 548, etc. 

(iv) Divisibility by 5: A number is divisible by 5 when its ones digit is 0 or 5. Example: 645, 540 etc.

(v) Divisibility by 6: A number is divisible by 6 when it is divisible by both 2 and 3. Example: 246, 7230, etc.

(vi) Divisibility by 9: A number is divisible by 9 when the sum of its digits is divisible by 9. Example: consider a number 215847. Sum of digits = 2+1+5+8+4+7 = 27 which is divisible by 9. Therefore, 215847 is divisible by 9. 

(vii) Divisibility by 10: A number is divisible by 10 when its ones digit is 0. Example: 540, 890, etc. 

(viii) Divisibility by 11: A number is divisible by 11 when the difference of the sum of its digits in odd places and the sum of its digits in even places is either o or a multiple of 11.
Example: consider a number 462.

Sum of digits in odd places = 4+2 = 6
Sum of digits in even places = 6
Difference = 6-6=0, which is zero. So, the number is divisible by 11.