Question

Bài 1: Chứng minh rằng:

a, 5^5 - 5^4 + 5^3 chia hết cho 7.

b, 7^6 + 7^5 - 7^4 chia hết cho 11.

c, 10^9 + 10^8 + 10^7 chia hết cho 222.

d, 10^6 - 5^7 chia hết cho 59.

e, 3^n+2 - 2^n+2 + 3^n - 2^n chia hết cho 10 với n \(\in\) N*.

f, 81^7 - 27^9 - 9^13 chia hết cho 45.

Answer 1

a/ \(5^5-5^4+5^3=5^3\left(5^2-5+1\right)=5^3.21⋮7\left(đpcm\right)\)

b/ \(7^6+7^5-7^4=7^4\left(7^2+7-1\right)=7^4.55⋮11\left(đpcm\right)\)

c/ \(10^9+10^8+10^7=10^7.\left(10^2+10+1\right)=10^7.111=1110000⋮222\left(đpcm\right)\)

d/ \(10^6-5^7=2^6.5^6-5^7=5^6\left(2^6-5\right)=5^6.59\left(đpcm\right)\)

e/ \(3^{n+2}-2^{n+2}+3^n-2^n=3^n\left(3^2+1\right)-2^n\left(2^2+1\right)=3^n.10-2^n.5=3^n.10-2^{n-1}.10=10\left(3^n-2^{n-1}\right)⋮10\left(đpcm\right)\)

f/ \(81^7-27^9-9^{13}=3^{28}-3^{27}-3^{26}=3^{26}\left(3^2-3-1\right)=3^{26}.5=3^{24}.45⋮45\left(đpcm\right)\)

Answer 2

a) Ta có: 5⁵ - 5⁴ + 5³

= 5³(5² - 5 + 1)

= 5³ . 3 . 7 \(⋮\) 7 (đpcm)