Math 9 Chapter 1 Lesson 12: The sign is divisible by 3, by 9
1. Summary of theory Tóm
1.1. Opening comments.
Considering the number 378, we see \(378 = 3 . 100 + 7 . 70 + 8.\)
Can write \(378 = 3 . ( 99 + 1) + 7 . ( 9 + 1) + 8\)
\(= 3 . 99 + 3 + 7 . 9 + 7 + 8\)
\(= ( 3 + 7 + 8) + ( 3 . 11 . 9 + 7 . 9)\)
= (sum of digits) + (number divisible by 9).
Comment: Every number can be written as the sum of its digits plus a number divisible by 9.
1.2. The sign is divisible by 9.
According to the opening comment: \(378 = ( 3 + 7 +8) + \)(number divisible by 9)
\(=\) \(18 +\) (number is divisible by 9).
The number 378 is divisible by 9 because both terms are divisible by 9.
Similarly consider: \(253 = (2 + 5 + 3) +\) (number is divisible by 9).
\(= 10 +\) (number is divisible by 9).
The number 253 is not divisible by 9 because one term is not divisible by 9, the other term is divisible by 9.
Conclude: Numbers whose sum of digits is divisible by 9 are divisible by 9 and only those numbers are divisible by 9.
1.3. The sign is divisible by 3.
According to the opening comment: \(2031 = ( 2 + 0 + 3 + 1) +\)(number is divisible by 9).
\(= 6 + \) (number is divisible by 9).
\(= 6 + \) (number is divisible by 3).
The number 2031 is divisible by 3 because both terms are divisible by 3.
Similarly consider: \(3415 = (3 + 4 + 1 + 5) +\)(number is divisible by 9).
\(= 13 +\) (number is divisible by 9).
\(= 13 + \) (number is divisible by 3).
The number 3415 is not divisible by 3 because one term is not divisible by 3, the other term is divisible by 3.
Conclude:
Numbers whose sum of digits is divisible by 3 are divisible by 3 and only those numbers are divisible by 3.
2. Illustrated exercise
Question 1: Which of the following numbers is divisible by 9: 621; 738; 451.
Solution guide:
We see the Sum of the digits of \(621 = 6 + 2 + 1 = 9 \) \(\vdots \) \(9\).
Sum of digits of \(738 = 7 + 3 + 8 = 18\) \(\vdots\) \(9\).
Sum of digits of \(451 = 4 + 5 + 1 = 10\) \(\not\vdots\) \(9\).
Verse 2: \(\overline{5*7}\) divisible by 3, so * could be?
Solution guide:
total \(5 + * + 7\) \(\vdots\) \(3\), * can be 0; 3; 6; 9.
Question 3: Consider the total \(1251 + 375\) Is it divisible by 3?
Solution guide:
Sum of digits of \(1251 = 1 + 2 + 5 + 1 = 9\) \(\vdots\) \(3\)
Sum of digits of \(375 = 3 + 7 + 5 = 15\) \(\vdots\)
Should infer \(1251 + 375 \) \(\vdots\) \(3\)
3. Practice
3.1. Essay exercises
Question 1: Which of the following numbers is divisible by 9: 765; 126; 637.
Lesson 2: \(\overline{5*1}\) divisible by 3, so * could be?
Question 3: Consider the total \(2346+ 624\) Is it divisible by 3?
3.2. Multiple choice exercises Bài
Question 1: Which of the following numbers is divisible by 3: 3214, 6789, 1243, 9831
A. 3214, 6789
B. 1243, 9831
C. 6789, 9831
D. 3214, 9831
Verse 2: Choose the correct statement from the statements below:
A. Numbers that are divisible by 3 are divisible by 9
B. Numbers that are divisible by 9 are divisible by 3
C. Numbers ending in 3 are divisible by 3
D. Numbers ending in 3 or 9 are divisible by 9
Question 3: Find the value of * so that \(\overline {4*7} \) is divisible by 9
A. 3
B. 5
C. 7
D. 9
Question 4: Find two natural numbers a, b such that \(\overline {a3b} \) is divisible by 2, 3, 5, 9?
A. a = b =0
B. a = 6, b = 5
C. a = 3, b = 0
D. a= 6, b =0
Question 5: From 1 to 100 how many numbers are divisible by 3?
A. 30 numbers
B. 31 numbers
C. 32 digits
D. 33 digits
4. Conclusion
Through this lesson, the signs are divisible by 3, by 9, you need to complete some of the goals that the lesson gives, such as:

Recognize signs of divisibility by 3 and 9.

Do the relevant exercises.
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