Ta có : a chia 6 dư 2 => a - 2 chia hết cho 6 => a - 2 + 12 chia hết cho 6 => a + 10 chia hết cho 6

a chia 7 dư 4 => a - 4 chia hết cho 7 => a - 4 + 14 chia hết cho 7 => a + 10 chia hết cho 7

=> a + 10 chia hết cho 6 và 7

=. a + 10 thuộc BC ( 6 ; 7 )

Mà BCNN ( 6 ; 7 ) = 42

=> a + 10 thuộc B ( 42 ) = { 0 ; 42 ; ... }

=> a + 10 chia 42 dư 42

=> a chia 42 dư 32

Vậy số a chia cho 42 dư 32

\(\dfrac{4}{x}+\dfrac{2}{y}=1\) \(\left(x;y\ne0\right)\)

\(\Rightarrow2x+4y=xy\)

\(\Rightarrow2x-8+4y-xy=-8\)

\(\Rightarrow2\left(x-4\right)-y\left(x-4\right)=-8\)

\(\Rightarrow\left(x-4\right)\left(y-2\right)=8\)

\(\Rightarrow\left(x-4;y-2\right)\in U\left(8\right)=\left\{-1;1;-2;2;-4;4;-8;8\right\}\)

\(\Rightarrow\left(x;y\right)\in\left\{\left(3;-6\right);\left(5;10\right);\left(2;-2\right);\left(8;4\right);\left(-4;;1\right);\left(12;3\right)\right\}\)

Với mọi x;y;z ta có:

\(\left\{{}\begin{matrix}\left(x-2\right)^2\ge0\\\left(y-2\right)^2\ge0\\\left(z-2\right)^2\ge0\end{matrix}\right.\)

\(\Rightarrow\left(x-1\right)^2+\left(y-1\right)^2+\left(z-1\right)^2\ge0\)

\(\Rightarrow x^2+y^2+z^2\ge4\left(x+y+z\right)-12\) (1)

Đồng thời cũng có:

\(\left\{{}\begin{matrix}\left(x-y\right)^2\ge0\\\left(y-z\right)^2\ge0\\\left(z-x\right)^2\ge0\end{matrix}\right.\)

\(\Rightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0\)

\(\Rightarrow2\left(x^2+y^2+z^2\right)\ge2\left(xy+yz+zx\right)\) 

\(\Rightarrow4\left(x^2+y^2+z^2\right)\ge4\left(xy+yz+zx\right)\)(2)

Cộng vế (1) và (2):

\(\Rightarrow5\left(x^2+y^2+z^2\right)\ge4\left(x+y+z+xy+yz+zx\right)-12=4.18-12=60\)

\(\Rightarrow x^2+y^2+z^2\ge\dfrac{60}{5}=12\)

Dấu "=" xảy ra khi \(x=y=z=2\)

Ex. 11 :

1. take -> takes

2. never is -> is never

3. waking -> wakes

4. has been -> was

5. will go -> go

6. aren't -> isn't

7. interested on -> interested in

8. as well as -> as soon as

9. keen into -> keen on

10. is -> are

11. telling -> to tell

12. what happened -> what had happened

13. was keen about -> was keen on

14. probably will -> will probably

15. will -> will be

Ex.12 :

1. I'm keen on maths.

2. Unless you listen, you won't follow the teacher's lecture.

3. We are interested in walking in the rain

4. Should you ask me for the car, i will lend it to you.

5. John spends 30 minutes to walk to school.

6. She's fond of playing cards with her children at the weekend.

7. She's interested in taking photos of her children.

8. We don't have any time to prepare the speech.

9. I promise I will phone you next week.

10. I am fond of watching the goldfish in the tank.

11. My mon isn't passionate about preparing meals.

12. He fancies reading comic books.

13. I think not many people are into listening to music in their free time.

14. Should he studies hard, he will pass the exam.

15. If you go out in the rain, you will get wet.

.............

Với k = 1 ta có:

A =  6k + 5 = 6.1 + 5 = 11 (là số nguyên tố)

Vậy tồn tại số nguyên tố dạng: 6k + 5 (đpcm)

Here are the answers based on the passage:

1. Where is Michael's school?

   • Michael's school is in California.

2. How long does it take him to go to school?

   • It takes Michael about 15 minutes to go to school.

3. What time do his classes end?

   • His classes end at 11:15.

4. What does he do in the afternoon?

   • In the afternoon, Michael plays sports and learns music.

5. What time does he go to bed?

   • He and his family often go to bed at 10 o'clock.

1 It is in California

2 It takes him 15 minutes to go to school

3 They end at 11:15

4 He plays sports and learns music

5 He goes to bed at 10

1. Line 1: "In England, when the school children come to school, they first go to the cloakroom."  
   - Redundant word: "school" (appears twice in the line)

2. Line 2: "They are take off their coats and raincoats, their caps and hats, and then go to their classroom."  
   - Redundant word: "are" (the correct phrase should be "They take off their coats...")

3. Line 3: "Some of the students go for to the laboratories and workshop where they learn in physics, chemistry, and art."  
   - Redundant word: "for" (the correct phrase should be "go to the laboratories...")

4. Line 4: "When a students is on duty, he comes to school very early."  
   - Redundant word: "a" (should be "When a student is on duty...")

5. Line 5: "He or she has to open all the window, water in the flower and clean the blackboard, so all everything is ready."  
   - Redundant word: "all" (should be "so everything is ready.")

line 1: school (1)

line 2: are 

line 3: their

line 4: for

line 5: in

line 6: students => student

line 7: very 

line 8: in

line 9: all

line 10: to