Question

a) So sánh : 9¹⁰ và 8⁹ + 7⁹ + 6⁹ + 5⁹ + 4⁹ + 3⁹+ 2⁹ + 1⁹

^b) Chứng minh : (36³⁶ - 9¹⁰) chia hết cho 45

Answer 1

a)Đặt \(A=8^9+7^9+6^9+5^9+4^9+3^9+2^9+1^9\)

\(A< 8^9+8^9+8^9+8^9+8^9+8^9+8^9+8^9\)

\(A< 8\cdot8^9\)

\(A< 8^{10}< 9^{10}\)

\(\Rightarrow9^{10}>8^9+7^9+6^9+5^9+4^9+3^9+2^9+1^9\)

Answer 2

a) \(8^9+7^9+6^9+5^9+4^9+3^9+2^9+1^9\)

(8+7+6+5+4+3+2+1)⁹

36⁹

Vậy36⁹>9⁹

Answer 3

b)Ta có:\(36^{36}-9^{10}=\left(....6\right)-\left(9^2\right)^5=\left(....6\right)-81^5=\left(....6\right)-\left(....1\right)=\left(....5\right)⋮5\left(1\right)\)

Ta có:

\(36^{36}-9^{10}=\left(9\cdot4\right)^{36}-9^{10}=9^{36}\cdot4^{36}-9^{10}=9\left(9^{35}\cdot4^{36}-9^9\right)⋮9\left(2\right)\)

Từ (1) và (2)

\(\Rightarrow36^{36}-9^{10}⋮\) 5 và 9

Mà ƯCLN(5;9)=1

\(\Rightarrow36^{36}-9^{10}⋮5\cdot9=45\left(đpcm\right)\)