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                 \ 10-Nov-2025  Bharat Arora   Proof-8                                     Reader’s Sign _______________________ Date __________

                    example 6. 341 ÷ 9 = ?                            example 8. 63 ÷ 9 = ?

                    Solution.                                         Solution.  Quotient = 6 + 1 = 7
                                                                      Remainder = 6 + 3 – 9 = 0
                               9 ) 341 (
                                                                      Case 2: When a three-digit number  is
                    Dividend = 341
                                                                      divisible by 9
                    Quotient = 3 | (3 + 4)   = 37
                    Remainder = 3 + 4 + 1 = 8                                            H T O
                    All the above examples are true as long                               a b c
                    as the sum of the digits is less than 9.             Quotient = a | a + b + 1, where a is in

                    Now let us take the case of division by 9,           the tens place, a + b + 1 is in the ones
                    when the sum of the digits in the dividend           place.
                                                                         Remainder = a + b + c – 9
                    is greater than or equal to 9.
                    The two  cases  for two- and three-digit          example 9. Divide 225 by 9
                    dividends are illustrated below.                  Solution. Since 2 + 2 + 5 = 9 is divisible
                                                                      by 9 .
                    Let  us take some  examples  to  make  it         Quotient = 2 tens and 2 + 2 + 1 = 5 ones = 25
                    clear.
                                                                      Remainder = 2 + 2 + 5 – 9 = 0
                    Case 1: When a two-digit number is                example 10. 459 ÷ 9 = ?
                    divisible by 9                                    Solution.

                                         T O                          Quotient = 4 tens and 4 + 5 + 1 = 10 ones or
                                         a b                          1 ten = 50
                                                                      Remainder =  4 + 5 + 9 – 9 = 9
                      Quotient = a + 1, Remainder = a + b – 9         Since the remainder = 9, we need to

                                                                      subtract 9 again and 1 will in due course
                    example 7. 45 ÷ 9 = ?                             be added to the quotient
                    Solution. Quotient = 4 + 1 = 5                    Quotient = 50 + 1 = 51

                    Remainder = 4 + 5 – 9 = 0                         Remainder = 9 – 9 = 0

                        78       Mathematics 4

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