n⁸+3n⁴-4=(n⁴+4)(n⁴-1)=(n⁴+4)(n²+1)(n+1)(n-1) Если число n не кратно 5, то оно дает при делении на 5 остатки либо1, либо2, либо3, либо 4, это означает, что число можно представить как 5k+1 либо 5k+2 либо 5k+3 либо 5k+4. При n=5k+1 n-1=5k+1-1=5k - кратно 5. Множитель (n-1) делится на 5, значит все произведение делится на 5.
При n=5k+2 n²+1=(5k+2)²+1=25k²+20k+4+1=25k²+20k+5=5(5k²+4k+1) - кратно 5. Множитель (n²+1) делится на 5, значит все произведение делится на 5.
При n=5k+3 n²+1=(5k+3)²+1=25k²+15k+9+1=25k²+15k+10=5(5k²+3k+2) - кратно 5. Множитель (n²+1) делится на 5, значит все произведение делится на 5.
При n=5k+4 n+1=(5k+4)+1=5k+5=5(k+1) - кратно 5. Множитель (n+1) делится на 5, значит все произведение делится на 5.
Computer Games Computers have become a major part of our everyday life. People simply can’t live without them. They work and study with the help of computers, find information, and, of course, play different games. Speaking about computer games, I should say that they are gaining more and more popularity both among kids and adults. There are two points of view on this question. Some people claim that such games are harmful and pointless. They destroy our life balance. First of all, computer games are unhealthy for the eye sight. If we look at the monitor for a long time, it gets worse. Secondly, sitting glued to the same place can make us fatter. Another argument against computer games is that they contain lots of violent scenes. Thus, playing them can make our behavior more aggressive. And, finally, computer games steal our time. We start rarely seeing our friends and relatives. To keep the balance it is better to have some active hobbies too, such as going in for sports or playing in a music band. There are some people who support computer games. They say that this activity has many benefits. For example, it can develop a good memory because while we play, we try to remember different details. Other than that, it develops logical thinking and even language skills. Spending lots of time online can lead to virtual friendship with other players. And, of course, for many people it is one of the best ways of entertainment. At the end, I’d like to say that so many men so many minds. As for me, I sometimes enjoy playing my favourite games, but I try to spend less time in front of the computer. Или по короче My favourite computer game
My favourite computer game is Minecraft. It is an indie game. This game was released in 2011. In spite of pixel graphics the game is very interesting and exciting! In this game you can build your own house or a castle. Also, it is possible for you to dig holes. Roughly speaking, in this game you can do everything.
The game is played over a million players around the world! I recommend this game to people who enjoy choosing actions in the game.
Если число n не кратно 5, то оно дает при делении на 5 остатки либо1, либо2, либо3, либо 4, это означает, что число можно представить как
5k+1 либо 5k+2 либо 5k+3 либо 5k+4.
При n=5k+1
n-1=5k+1-1=5k - кратно 5.
Множитель (n-1) делится на 5, значит все произведение делится на 5.
При n=5k+2
n²+1=(5k+2)²+1=25k²+20k+4+1=25k²+20k+5=5(5k²+4k+1) - кратно 5.
Множитель (n²+1) делится на 5, значит все произведение делится на 5.
При n=5k+3
n²+1=(5k+3)²+1=25k²+15k+9+1=25k²+15k+10=5(5k²+3k+2) - кратно 5.
Множитель (n²+1) делится на 5, значит все произведение делится на 5.
При n=5k+4
n+1=(5k+4)+1=5k+5=5(k+1) - кратно 5.
Множитель (n+1) делится на 5, значит все произведение делится на 5.
Перевод на русский язык
Computer Games
Computers have become a major part of our everyday life. People simply can’t live without them. They work and study with the help of computers, find information, and, of course, play different games. Speaking about computer games, I should say that they are gaining more and more popularity both among kids and adults. There are two points of view on this question. Some people claim that such games are harmful and pointless. They destroy our life balance. First of all, computer games are unhealthy for the eye sight. If we look at the monitor for a long time, it gets worse. Secondly, sitting glued to the same place can make us fatter. Another argument against computer games is that they contain lots of violent scenes. Thus, playing them can make our behavior more aggressive. And, finally, computer games steal our time. We start rarely seeing our friends and relatives. To keep the balance it is better to have some active hobbies too, such as going in for sports or playing in a music band. There are some people who support computer games. They say that this activity has many benefits. For example, it can develop a good memory because while we play, we try to remember different details. Other than that, it develops logical thinking and even language skills. Spending lots of time online can lead to virtual friendship with other players. And, of course, for many people it is one of the best ways of entertainment. At the end, I’d like to say that so many men so many minds. As for me, I sometimes enjoy playing my favourite games, but I try to spend less time in front of the computer.
Или по короче
My favourite computer game
My favourite computer game is Minecraft. It is an indie game. This game was released in 2011. In spite of pixel graphics the game is very interesting and exciting! In this game you can build your own house or a castle. Also, it is possible for you to dig holes. Roughly speaking, in this game you can do everything.
The game is played over a million players around the world! I recommend this game to people who enjoy choosing actions in the game.