a) Tìm số nguyên x,y biết: ( x - 3 ) . ( y + 1 ) = 15b) Cho M = 1+ 3+32 + 33 + 34 + …+ 399 + 3100 . Tìm số dư khi chia M cho 13, chia M cho 40...
Đọc tiếp
a) Tìm số nguyên x,y biết: ( x - 3 ) . ( y + 1 ) = 15
b) Cho M = 1+ 3+32 + 33 + 34 + …+ 399 + 3100 . Tìm số dư khi chia M cho 13, chia M cho 40 .
Answer:
a)
Có:
\(15=1.15=\left(-1\right).\left(-15\right)=3.5=\left(-3\right).\left(-5\right)\)
b)
Có:
\(M=1+3+3^2+3^3+3^4+...+3^{98}+3^{99}+3^{100}\)
\(=\left(1+3\right)+\left(3^2+3^3+3^4\right)+...+\left(3^{98}+3^{99}+3^{100}\right)\)
\(=4+3^2.\left(1+3+3^2\right)+...+3^{98}.\left(1+3+3^2\right)\)
\(=4+3^2.13+3^{98}.13\)
\(=4+13.\left(3^2+...+3^{98}\right)\)
Vậy M chia 13 dư 4
Có:
\(M=1+3+3^2+3^3+3^4+...+3^{99}+3^{100}\)
\(=1+\left(3+3^2+3^3+3^4\right)+\left(3^5+3^6+3^7+3^8\right)+...+\left(3^{97}+3^{98}+3^{99}+3^{100}\right)\)
\(=1+3.40+3^5.40+...+3^{97}.40\)
\(=1+40.\left(3+3^5+...+3^{97}\right)⋮40\)
Mà \(40\left(3+3^5+...+3^{97}\right)⋮40\)
Vậy M chia 40 dư 1