Question 9
Polynomials and Multiples of 9
It is easy to find the multiples of 9 from 1 to 100. But can you easily find the multiples of 9 for numbers larger than 100 or even 10 000 without using a calculator?
In order to determine whether a number is a multiple of 9, we can add up all digits in the number. If the sum is divisible by 9, then the number is a multiple of 9.
e.g. (i) Consider 171. Sum of all digits = 1+7+1= 9 , so 171 is a multiple of 9.
(ii) Consider 123 123 123. Sum of all digits = 1 +2 +3 +1 +2 +3 +1 +2 +3 =18 since 18 is a multiple of 9. so 123123123 is a multiple of 9.
(iii) Consider 100 001. Sum of all digits = 1+0+0+0+0+1= 2
since 2 is not a multiple of 9.
so 100 001 is not a multiple of 9.
In fact, we can use the knowledge of polynomials to prove that the above method works.
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