已知3^1=3,3^2=9,3^3=27,3^4=81……求(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2的个位数字.
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![已知3^1=3,3^2=9,3^3=27,3^4=81……求(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2的个位数字.](/uploads/image/z/1026783-63-3.jpg?t=%E5%B7%B2%E7%9F%A53%5E1%3D3%2C3%5E2%3D9%2C3%5E3%3D27%2C3%5E4%3D81%E2%80%A6%E2%80%A6%E6%B1%82%283-1%29%283%2B1%29%283%5E2%2B1%29%283%5E4%2B1%29%E2%80%A6%E2%80%A6%283%5E32%2B1%29%2B2%E7%9A%84%E4%B8%AA%E4%BD%8D%E6%95%B0%E5%AD%97.)
已知3^1=3,3^2=9,3^3=27,3^4=81……求(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2的个位数字.
已知3^1=3,3^2=9,3^3=27,3^4=81……求(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2的个位数字.
已知3^1=3,3^2=9,3^3=27,3^4=81……求(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2的个位数字.
(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2
=(3^2-1)(3^2+1)(3^4+1)……(3^32+1)+2
=(3^4-1)(3^4+1)……(3^32+1)+2
=...
=(3^32-1)(3^32+1)+2
=(3^64-1)+2
=3^64+1
因为
3^4=81个位是1
所以
3^64=(3^4)^16个位也是1,所以
原式的个位是1+1=2
(3-1)(3+1)(3^2+1)(3^4+1)……(3^32+1)+2
=(3^2-1)(3^2+1)(3^4+1)……(3^32+1)+2
=(3^4-1)(3^4+1)……(3^32+1)+2
=...
=(3^32-1)(3^32+1)+2
=(3^64-1)+2
=3^64+1
因为
3的次方每4次方个位均为1,64次方是4的整数倍,故个位为1,
所以
原式的个位是1+2=3