-2.

-1.

0.

1.

2.

3.

4.

5.

6

=0

hmm

mk ko bit

bn thông cảm lúc mik dán câu hỏi vào thì bị thế mà k để ý cứ thế gửi

Lời giải:

Gọi biểu thức là $A$. Đặt $n=2k+1$ với $k$ nguyên.

$A=n^8(n^4-1)-(n^4-1)=(n^4-1)(n^8-1)$

$=(n^4-1)(n^4-1)(n^4+1)$

$=(n-1)^2(n+1)^2(n^2+1)^2(n^4+1)$

$=(2k)^2(2k+2)^2(4k^2+4k+2)^2(n^4+1)$

$=64[k(k+1)]^2(2k^2+2k+1)^2(n^4+1)$

Vì $k(k+1)$ là tích 2 số nguyên liên tiếp nên hiển nhiên chia hết cho 2

$\Rightarrow [k(k+1)]^2\vdots 4$

Với $n$ lẻ thì hiển nhiên $n^4+1\vdots 2$

$\Rightarrow A\vdots 64.4.2=512$ (đpcm)

\(1152=32.36\)

Đặt \(A=n^8-n^6-n^4+n^2=n^6\left(n^2-1\right)-n^2\left(n^2-1\right)\)

\(=n^2\left(n^2-1\right)\left(n^4-1\right)=n^2\left(n^2-1\right)\left(n^2-1\right)\left(n^2+1\right)\)

\(=\left[n\left(n-1\right)\left(n+1\right)\right]^2\left(n^2+1\right)\)

Do \(n\) lẻ \(\Rightarrow n=2k+1\)

\(\Rightarrow A=\left[\left(2k+1\right)\left(2k+1-1\right)\left(2k+1+1\right)\right]^2\left[\left(2k+1\right)^2+1\right]\)

\(=32\left[k\left(k+1\right)\left(2k+1\right)\right]^2.\left(2k^2+2k+1\right)\)

Do \(k\) và k+1 là 2 số tự nhiên liên tiếp \(\Rightarrow k\left(k+1\right)⋮2\) (1)

Nếu k chia hết cho 3 \(\Rightarrow k\left(k+1\right)\left(2k+1\right)⋮3\)

Nếu k chia 3 dư 1 \(\Rightarrow2k+1⋮3\Rightarrow k\left(k+1\right)\left(2k+1\right)⋮3\)

Nếu k chia 3 dư 2 \(\Rightarrow k+1⋮3\Rightarrow k\left(k+1\right)\left(2k+1\right)⋮3\)

\(\Rightarrow k\left(k+1\right)\left(2k+1\right)\) luôn chia hết cho 3 (2)

(1);(2) \(\Rightarrow k\left(k+1\right)\left(2k+1\right)⋮6\Rightarrow\left[k\left(k+1\right)\left(2k+1\right)\right]^2⋮36\)

\(\Rightarrow32\left[k\left(k+1\right)\left(2k+1\right)\right]^2⋮\left(32.36\right)\Rightarrow A⋮1152\)

ảnh đại diện trên google kìa

Đặt: \(A=n^8-n^6-n^4+n^2\)

\(A=\left(n^8-n^6\right)-\left(n^4-n^2\right)\)

\(A=n^6\left(n^2-1\right)-n^2\left(n^2-1\right)\)

\(A=\left(n^2-1\right)\left(n^6-n^2\right)\)

\(A=\left(n-1\right)\left(n+1\right)n^2\left(n^4-1\right)\)

\(A=n^2\left(n-1\right)\left(n+1\right)\left[\left(n^2\right)^2-1\right]\)

\(A=n^2\left(n-1\right)\left(n+1\right)\left(n^2-1\right)\left(n^2+1\right)\)

\(A=n^2\left(n-1\right)\left(n+1\right)\left(n-1\right)\left(n+1\right)\left(n^2+1\right)\)

\(A=n\left(n-1\right)\left(n+1\right)n\left(n-1\right)\left(n+1\right)\left(n^2+1\right)\)

Ta có: \(n\left(n-1\right)\left(n+1\right)\) là tích của 3 số tự nhiên liên tiếp nên sẽ chia hết cho 3 

Còn: \(\left[n\left(n-1\right)\left(n+1\right)\right]\left[n\left(n-1\right)\left(n+1\right)\right]\) sẽ chia hết cho \(3\times3=9\) 

Do n sẽ là số lẻ nên \(\left(n-1\right);\left(n+1\right)\) sẽ luôn luôn là số chẵn 

Mà: \(\left(n-1\right)\left(n+1\right)\) sẽ chia hết cho 8 vì tích của hai số chẵn liên liếp sẽ chia hết cho 8 

Còn  \(\left(n+1\right)\left(n-1\right)\left(n+1\right)\left(n-1\right)\left(n^2+1\right)\) sẽ chia hết cho \(8\cdot8\cdot2=128\) 

Ta có: 

\(\text{Ư}\text{C}LN\left(9;128\right)=1\)

Nên: A ⋮ \(9\cdot128=1152\left(dpcm\right)\)

`n^8+4n^7+6n^6+4n^5+n^4=n^4(n^4+4n^3+6n^2+4n+1)=n^4(n+1)^4=(n(n+1))^4=(2k)^4=16k^2\vdots16` với `k\inNN`

Mà \(125⋮5\Rightarrow\left(2n-1\right)^3+75⋮5\) mà \(75⋮5\Rightarrow\left(2n-1\right)^3⋮5\)

Vì 5 nguyên tố \(\Rightarrow2n-1⋮5\Rightarrow\left(2n-1\right)^3⋮125\) nhưng 75 \(⋮̸\)125 (vô lí)

Vậy \(4n^3-6n^2+3n+37\)\(⋮̸\)125

.

MORE EXERCISE

\n\n

Word Form :
\n1. This book is very ___heavy_. (heavily)
\n2. The man works ___heavily__ all day. (heavy)
\n3. I don’t like the ___noise___ of vehicles on the street. (noisy)
\n4. The music is very ___noisy____. I can’t study. (noise)
\n5. These boys always do things ___noisily___. (noisy)
\n6. That man is really __strong__. (strength)
\n7. It will take you a few weeks to build up your ___strength_ again. (strong)
\n8. What is your ___favorite____ food, Mai? (favor)
\n9. That room is decorated with many ____colorfully_. (colorful)
\n10. How many ___color_ are there in this flag? (colorful)
\n11. A young girl is _colorful__ the picture in her room at present. (color)
\n12. We should eat fruit to stay ___healthy__. (health)
\n13. His sisters are __gymnast___. They’re strong. (gym)
\n14. Her ___favorite__ drink is fruit juice. (favor)
\n15. Vietnamese people are really ____friendly__. (friend)
\n16. Minh and Nam are my close _friend_____. (friendly)
\n17. We don’t like him because he’s __unfriendly___. (friend)
\n18. Milk and fruit juice are ___healthy___ drinks. (health)
\n19. The old people live ___healthy__ in the countryside.
\n20. This is a busy shopping mall, full of activities and ___color____. (colorful)

cám ơn bạn

Hãy đổi những câu sau ra lời tường thuật
\n1. He said to them, "Don't use bent coins in a slot machine"
\n-> He told them not to use bent coins in a slot machine.
\n2. She said to him, "Remember to send this message to my father"
\n-> She reminded him to send this message to his father.
\n3. "Write to me as often as you can," said his wife
\n-> His wife told me to write to her as often as I could.
\n4. They said, "Our father has just driven to his office"
\n-> They said that their father had just driven to his office.
\n5. She said to him, "You don't remember borrowing my book yesterday"
\n-> She reminded him to borrow his book the previous day.
\n6. They said to her, "We will help you if you want"
\n-> They told her that they would help her if she wanted.
\n7. He said to me, "If you think the room is hot, you can open the window"
\n-> He told me that if I thought the room was hot, I could open the window.
\n8. She said to the boy, "Do you like to play football or volleyball?"
\n-> He asked the boy if he liked to play football or volleyball.
\n9. He said to them, "Did you go to the beach last summer?"
\n-> He asked them if they had gone to the beach the previous summer.
\n10. He said to her, "Whose motorbike did you ride yesterday?"
\n-> He. asked me whose motorbike I had driven the previous day.