## PLAYING WITH NUMBERS - ESSENTIAL POINTS

• Number in general form: A number 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.
• Tests of Divisibility:
• 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.
• 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.
• 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.
• Divisibility by 5: A number is divisible by 5 when its ones digit is 0 or 5. Example: 645, 540 etc.
• Divisibility by 6: A number is divisible by 6 when it is divisible by both 2 and 3. Example: 246, 7230, etc.
• Divisibility by 9: A number to be divisible by 9 has 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.
• Divisibility by 10: A number is divisible by 10 when its ones digit is 0. Example: 540, 890, etc.
• 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
• (10n - 1) is divisible by 11 for even values of n.
• The largest natural number by which the product of three consecutive even natural numbers is always divisible, is 48.
• If a number is divisible by any number m, then it will also be divisible by each of the factor of m.