Вместо Х подставим (4) 5*4+17=20+17=37
Заметим: ![6^{n}-1\equiv 1^{n}-1(mod5)=1-1=0,n \in N_0=>(6^{n}-1)\vdots 5=>2*(6^{n}-1)\vdots 10=>6*(6^{n}-1)\vdots 10=>(6^{n+1}-6)\vdots 10=>6^{n+1}\equiv6(mod 10)](https://tex.z-dn.net/?f=6%5E%7Bn%7D-1%5Cequiv%201%5E%7Bn%7D-1%28mod5%29%3D1-1%3D0%2Cn%20%5Cin%20N_0%3D%3E%286%5E%7Bn%7D-1%29%5Cvdots%205%3D%3E2%2A%286%5E%7Bn%7D-1%29%5Cvdots%2010%3D%3E6%2A%286%5E%7Bn%7D-1%29%5Cvdots%2010%3D%3E%286%5E%7Bn%2B1%7D-6%29%5Cvdots%2010%3D%3E6%5E%7Bn%2B1%7D%5Cequiv6%28mod%2010%29)
![1)\:2017^{2018}+2018^{2019} - 117^{78}\equiv 7^{2018}+8^{2019} - 7^{78}(mod 10)=7^{78}*(7^{1940}-1)+8^{2019} =49^{39}*(49^{970}-1)+8*64^{1009}\equiv (-1)^{39}*((-1)^{970}-1)+8*4^{1009}(mod10)=(-1)*0+32*256^{252}\equiv 2*6^{252}(mod10)\equiv 2*6(mod10)=12\equiv 2(mod 10)](https://tex.z-dn.net/?f=1%29%5C%3A2017%5E%7B2018%7D%2B2018%5E%7B2019%7D%20-%20117%5E%7B78%7D%5Cequiv%207%5E%7B2018%7D%2B8%5E%7B2019%7D%20-%207%5E%7B78%7D%28mod%2010%29%3D7%5E%7B78%7D%2A%287%5E%7B1940%7D-1%29%2B8%5E%7B2019%7D%20%3D49%5E%7B39%7D%2A%2849%5E%7B970%7D-1%29%2B8%2A64%5E%7B1009%7D%5Cequiv%20%28-1%29%5E%7B39%7D%2A%28%28-1%29%5E%7B970%7D-1%29%2B8%2A4%5E%7B1009%7D%28mod10%29%3D%28-1%29%2A0%2B32%2A256%5E%7B252%7D%5Cequiv%202%2A6%5E%7B252%7D%28mod10%29%5Cequiv%202%2A6%28mod10%29%3D12%5Cequiv%202%28mod%2010%29)
![2)\:28^{30}*42^{47}+19^{2648}\equiv (-2)^{30}*2^{47}+(-1)^{2648}(mod10)=2^{77}+1=4*32^{15}+1\equiv 4*2^{15}+1(mod 10)=4*32^3+1\equiv 4*2^3+1(mod10)=4*8+1=33\equiv 3(mod10)](https://tex.z-dn.net/?f=2%29%5C%3A28%5E%7B30%7D%2A42%5E%7B47%7D%2B19%5E%7B2648%7D%5Cequiv%20%28-2%29%5E%7B30%7D%2A2%5E%7B47%7D%2B%28-1%29%5E%7B2648%7D%28mod10%29%3D2%5E%7B77%7D%2B1%3D4%2A32%5E%7B15%7D%2B1%5Cequiv%204%2A2%5E%7B15%7D%2B1%28mod%2010%29%3D4%2A32%5E3%2B1%5Cequiv%204%2A2%5E3%2B1%28mod10%29%3D4%2A8%2B1%3D33%5Cequiv%203%28mod10%29)
3) ![6^{127}:2=6^{126}*3\equiv6*3(mod10)=18\equiv 8(mod10)](https://tex.z-dn.net/?f=6%5E%7B127%7D%3A2%3D6%5E%7B126%7D%2A3%5Cequiv6%2A3%28mod10%29%3D18%5Cequiv%208%28mod10%29)
Ответ: 1)2; 2)3; 3)8
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Использованы свойства сравнения чисел по модулю
4*(220+4)=4*24=96
5*(10+2)5*12=60
90:3+3:3=30+1=31
40:4+12:4=10+3=13
(90+3):3=93:3=31
4*20+4*4=80+16=96
(40+12):4=52:4=13
5*10+5*2=50+10=60
5 тонн он весит или 5000 килограмм