Divisibility by 9

### 1.1. Knowledge to remember

*a) Example*

- 72 : 9 = 8. We have : 7 + 2 = 9, 9 : 9 = 1.
- 657 : 9 = 73. We have : 6 + 5 + 7 = 18, 18 : 9 = 2.
- 182 : 9 = 20 (remainder 2). We have: 1 + 8 + 2 = 11, 11 : 9 = 1 (remainder 2).
- 451 : 9 = 50 (remainder 1). We have : 4 + 5 + 1 = 10, 10 : 9 = 1 (remainder 1)

*b) The sign is divisible by 9*

*Numbers whose sum of digits is divisible by 9 is divisible by 9.*

*Attention :* Numbers whose sum of digits is not divisible by 9 is not divisible by 9.

### 1.2. Solving Textbook Exercises

**Lesson 1: **Which of the following numbers is divisible by 9?

99 ; 1999 ; 108 ; 5643 ; 29385

**Solution guide:**

- Sum the digits of each number, considering whether the sum is divisible by 9 or not.
- Numbers whose sum of digits is divisible by 9 is divisible by 9.

+ The number 99 has the sum of digits 9 + 9 = 18.

Since 18 is divisible by 9, so 99 is divisible by 9.

+ The number 1999 has the sum of digits 1 + 9 + 9 + 9 = 28.

Since 28 is not divisible by 9, so 1999 is not divisible by 9.

+ The number 108 has the sum of digits 1 + 0 + 8 = 9.

Since 9 is divisible by 9, so 108 is divisible by 9.

+ The number 5643 has the sum of digits 5 + 6 + 4 + 3 = 18.

Since 18 is divisible by 9, the number 5643 is divisible by 9.

+ The number 29385 has the sum of digits 2 + 9 + 3 + 8 + 5 = 27.

Since 27 is divisible by 9, the number 29385 is divisible by 9.

So out of the given numbers, the numbers that are divisible by 9 are:

99 ; 108 ; 5643 ; 29385

**Lesson 2: **Which of the following numbers is not divisible by 9?

96 ; 108 ; 7853 ; 5554 ; 1097

__Solution guide:__

- Sum the digits of each number, considering whether the sum is divisible by 9 or not.
- Numbers whose sum of digits is not divisible by 9 is not divisible by 9.

+ The number 96 has the sum of digits 9 + 6 = 15.

Since 15 is not divisible by 9, so 96 is not divisible by 9.

+ The number 108 has the sum of digits 1 + 0 + 8 = 9.

Since 9 is divisible by 9, so 108 is divisible by 9.

+ Number 7853 has the sum of digits 7 + 8 + 5 + 3 = 23.

Since 23 is not divisible by 9, so 7853 is not divisible by 9.

+ The number 5554 has the sum of digits 5 + 5 + 5 + 4 = 19.

Since 19 is not divisible by 9, so 5554 is not divisible by 9.

+ The number 1097 has the sum of digits 1 + 0 + 9 + 7 = 17.

Since 17 is not divisible by 9, the number 1097 is not divisible by 9.

So among the given numbers, the numbers that are not divisible by 9 are:

96 ; 7853 ; 5554 ; 1097.

**Lesson 3:** Write two three-digit numbers that are divisible by 9

__Solution guide:__

- Based on the sign of divisibility by 9: Numbers whose sum of digits is divisible by 9 is divisible by 9.

Students can write the following: 351; 684.

**Lesson 4: **Find the correct number to write in the blank to get the number divisible by 9

__Solution guide:__

- Based on the sign of divisibility by 9: Numbers whose sum of digits is divisible by 9 is divisible by 9.

Suppose the digit to fill in the blank is x.

+ If the base \(\overline {31x} \) is divisible by 9, then the sum of the digits is divisible by 9, or 3+1+x = 4+x is divisible by 9. So x = 5.

+ If the base \(\overline {x35} \) is divisible by 9, then the sum of the digits is divisible by 9, or x+3+5 = x+8 is divisible by 9. So x = 1.

+ If the base \(\overline {2×5} \) is divisible by 9, then the sum of the digits is divisible by 9, or 2+x+5 = 7+x is divisible by 9. Hence x = 2.

We have the following result:

.

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