Question

[Đỗ Thị Ánh]

Chứng minh rằng với mọi số tự nhiên nguyên dương n thì :

a ) 10⁹ + 10⁸ + 10⁷ \(⋮\) 222

b) 81⁷ - 27⁹ - 9¹³ \(⋮\) 45

Answer 1

[Nguyễn Phương Anh]

a, \(10^9+10^8+10^7⋮222\)

Ta có:\(10^9+10^8+10^7=10^7.\left(10^2+10+1\right)\)

\(=10^7.111=5^7.2^7.111=5^7.2^6.2.111=5^7.2^6.222\)

Vì 222\(⋮222\Rightarrow5^7.2^6.222⋮222\)

Vậy \(10^9+10^8+10^7⋮222\)

b) 81⁷ - 27⁹ - 9¹³ 45

\(\)Ta có: \(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)

\(=3^{28}-3^{27}-3^{26}=3^{26}.\left(3^2-3-1\right)\)

\(=3^{26}.5=3^{24}.3^2.5=3^{24}.45\)

Vì \(45⋮45\Rightarrow3^{24}.45⋮45\)

Vậy \(81^7-27^9-9^{13}⋮45\)

CHÚC BẠN HỌC TỐT!!

Answer 2

[An Trần]

a) \(10^9+10^8+10^7\)

\(=10^7\left(10^2+10+1\right)\)

\(=5^7.2^7.\left(100+10+1\right)\)

\(=5^7.2^6.2\left(100+10+1\right)\)

\(=5^7.2^6.2.111\)

\(=5^7.2^6.222⋮222\)