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Tests for Divisibility of Numbers - Divisibility by 9

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Divisibility by 9:

If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.

5283 is divisible by 9 (i.e., 5 + 2 + 8 + 3 = 18).
Is the number 758 divisible by 9? No. The sum of its digits 7 + 5 + 8 = 20 is also not divisible by 9.

More Examples:

Is the number 57225834 divisible by 9?
Now, sum the digits of the number,
5 + 7 + 2 + 2 + 5 + 8 + 3 + 4 = 36
36 is divisible by 9, so 57225834 is divisible by 9.

Is the number 3216763132 divisible by 3?
3 + 2 + 1 + 6 + 7 + 6 + 3 + 1 + 3 + 2 = 34
34 is not divisible by 9, so the number 3216763132 is not divisible by 9.

Let us take some number say 3573.
Its expanded form is: 3 × 1000 + 5 × 100 + 7 × 10 + 3
This is equal to 3 × (999 + 1) + 5 × (99 + 1) + 7 × (9 + 1) + 3
= 3 × 999 + 5 × 99 + 7 × 9 + (3 + 5 + 7 + 3)
We see that the number 3573 will be divisible by 9 if (3 + 5 + 7 + 3) is divisible by 9. We see that 3 + 5 + 7 + 3 = 18 is divisible by 9.
Therefore, the number 3573 is divisible by 9.

Let us explain divisible by 9 rule:

If the number is 'cba', then, 100c + 10b + a = 99c + 9b + (a + b + c)

                                                                       = `ubrace(9(11c + b))_("divisible by 9") + (a + b + c)`

Hence, divisibility by 9 is possible if a + b + c is divisible by 9.

Example

Check the divisibility of 21436587 by 9.
The sum of the digits of 21436587 is 2 + 1 + 4 + 3 + 6 + 5 + 8 + 7 = 36.
This number is divisible by 9 (for 36 ÷ 9 = 4).
We conclude that 21436587 is divisible by 9.

Example

Check the divisibility of 152875 by 9.

The sum of the digits of 152875 is 1 + 5 + 2 + 8 + 7 + 5 = 28.
This number is not divisible by 9.
We conclude that 152875 is not divisible by 9.

Example

If the three-digit number 24x is divisible by 9, what is the value of x?
Since 24x is divisible by 9, sum of its digits, i.e., 2 + 4 + x should be divisible by 9, i.e., 6 + x should be divisible by 9.
This is possible when 6 + x = 9 or 18, ....
But, since x is a digit, therefore, 6 + x = 9, i.e., x = 3.
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