16^x < 128⁴

(2⁴)^x < (2⁷)⁴

2^4x < 2²⁸

4x < 28

  x < 7

xϵ { 0; 1; 2; 3; 4;5;6}

SSH: (101-1)/1 +1= 100

Tổng=(100+1)*100/2=5050

CM (9ⁿ + 1)(9ⁿ + 2)(9ⁿ+ 3)(9ⁿ+4) ⋮ 5

nếu n ⋮ 2 ⇔ n = 2k ⇔ 9ⁿ + 4= 9^2k+4  = (9²)^k + 4 = 81^k+4 = \(\overline{...5}\) ⋮ 5

⇔ (9ⁿ + 1)(9ⁿ + 2)(9ⁿ+ 3)(9ⁿ+4) ⋮ 5 (1)

nếu n : 2 dư 1⇔ n = 2k+1  ta có :

9ⁿ+1 = 9^2k+1+1 ⇔(9²)^k.9 +1= 81^k.9 + 1=  \(\overline{...9}\) + 1 = \(\overline{...0}\) ⋮ 5

(9ⁿ + 1)(9ⁿ + 2)(9ⁿ+ 3)(9ⁿ+4) ⋮ 5 (2)

kết hợp (1) và(2) ta có:

(9ⁿ + 1)(9ⁿ + 2)(9ⁿ+ 3)(9ⁿ+4) ⋮ 5 ∀ n ϵ N (đpcm)

Giả sử a>b>c>d

Số lớn nhất là: \(\overline{abcd}\)

Số nhỏ nhất là \(\overline{dcba}\)

Theo đề bài

\(\overline{abcd}+\overline{dcba}=11330\)

\(1000a+100b+10c+d+1000d+100c+10b+a=11330\)

\(1001a+110b+110c+1001d=11330\)

\(91a+10b+10c+91d=1030\)

91(a+d)+10(b+c)=1030

Ta có

\(1030⋮10;10\left(b+c\right)⋮10\Rightarrow91\left(a+d\right)⋮10\Rightarrow a+d=10\)

=> 91.10+10(b+c)=1030 => b+c=12

\(\Rightarrow a+b+c+d=10+12=22\)

\(12\left(x-1\right):3=72\\ =>12\left(x-1\right)=72\times3\\ =>12\left(x-1\right)=216\\ =>\left(x-1\right)=18\\ =>x-1=18\\ =>x=19\)

12(x-1):3 = 72

12(x-1) = 72 . 3

12(x-1) = 216

x-1 = 216 : 12

x-1 = 18

x = 18+1

x = 19

\(1^3+2^3+3^3=1+8+27=36\)

1+8+27=36

2

3⁴¹ > 4³¹

3^41<4^31

\(3^x+3^{x+1}=324\\ =>3^x+3^x.3^1=324\\ =>3^x+3^x.3=324\\ =>3^x\left(1+3\right)=324\\ =>3^x.4=324\\ =>3^x=324:4\\ =>3^x=81\\ =>3^x=3^4\\ =>x=4\)

3x + 3x+1 = 324

3x + 3x.3 = 324

3x.(1 + 3) = 324

3x.4 = 324

3x = 324 : 4

3x = 81

3x = 34

<=> x = 4

65¹²⁵               < 98⁶²⁵

Ta có : 98 > 65 ; 125 > 625 nên 65¹²⁵ > 98⁶²⁵