\(A=11^3+12^3+...+1945^3\)

Ta có: \(A=11^3+12^3+...+1945^3\)

\(=\left(12^3+14^3+...+1944^3\right)+\left(11^3+13^3+...+1945^3\right)\)

Do dãy \(11;13;...;1945\) có \(\frac{1945-11}{2}+1=968\) số hạng

\(\Rightarrow \left(11^3+13^3+...+1945^3\right)⋮2\) mà \(\left(12^3+14^3+...+1944^3\right)⋮2\)

\(\Rightarrow A⋮2\left(1\right)\)

Mặt khác:

\(A=\left(11^3+1945^3\right)+\left(12^3+1944^3\right)+...+\left(977^3+979^3\right)+978^3\)

\(=967.1956^3+978^3⋮3\)

\(\Rightarrow A⋮3\left(2\right)\)

Từ \(\left(1\right);\left(2\right)\Rightarrow A⋮6\)

a) \(4^{13}+4^{14}+4^{15}+4^{16}=4^{13}\left(1+4\right)+4^{14}\left(1+4\right)=4^{13}.5+4^{14}.5=5\left(4^{13}+4^{14}\right)⋮5\Rightarrow dpcm\)

c) \(2^{10}+2^{11}+2^{12}+2^{13}+2^{14}+2^{15}\)

\(=2^{10}\left(1+2+2^2\right)+2^{13}\left(1+2+2^2\right)\)

\(=2^{10}.7+2^{13}.7=7\left(2^{10}+2^{13}\right)⋮7\Rightarrow dpcm\)

Câu c bạn xem lại đê

\(M=1+2+3+4+5+6+7+8+9+10+11+12+13+14\)

\(=\left(1+14\right)+\left(2+13\right)+\left(3+12\right)+...+\left(6+9\right)+\left(7+8\right)\)

\(=15+15+15+...+15+15\)

\(=15\times7=105\)

\(1+3+5+7+9+11+13\)

\(=\left(1+13\right)+\left(3+11\right)+\left(5+9\right)+7\)

\(=14+14+14+7=49\)

Ta có:    \(105\div49=2\)dư   \(7\)

Vậy    \(M\)ko chia hết cho   \(1+3+5+7+9+11+13\)

yes or no

ko chia hết.Vì 1+2+3+.......+13 \(⋮\) 1+2+....+13 mà 14 ko\(⋮\) cho 1+2+.......+13

a, 11 + 11² + 11³ + ... + 11⁷ + 11⁸

= (11 + 11²) + (11³ + 11⁴) + ... + (11⁷ + 11⁸)

= 11(1 + 11) + 11³(1 + 11) + ... + 11⁷(1 + 11)

= 11.12 + 11³.12 + .... + 11⁷.12

= 12(11 + 11³ + ... + 11⁷) chia hết cho 12

b, 7 + 7² + 7³ + 7⁴

= (7 + 7³) + (7² + 7⁴)

= 7(1 + 7²) + 7²(1 + 7²)

= 7.50 + 7².50

= 50(7  + 7²) chia hết cho 50

c, 3 + 3² + 3³ + 3⁴ + 3⁵ + 3⁶

= (3 + 3² + 3³) + (3⁴ + 3⁵ + 3⁶)

= 3(1 + 3 + 3²) + 3⁴(1 + 3 + 3²)

= 3.13 + 3⁴.13

= 13(3 + 3⁴) chia hết cho 13

Ta có :

A = 13! - 11! = 11! . 12 . 13 - 11! = 11! . (12 . 13 - 1) = 11! . 155 chia hết cho 155

b)=3^1+(3^2+3^3+3^4)+(3^5+3^6+3^7)+....+(3^58+3^59+3^60)

=3^1+(3^2.1+3^2.3+3^2.9)+(3^5.1+3^5.3+3^5.9)+......+(3^58.1+3^58.3+3^58.9)

=3^1+3^2.(1+3+9)+3^5.(1+3+9)+.....+3^58.(1+3+9)

=3+3^2.13+3^5.13+.........+3^58.13

=3.13.(3^2+3^5+....+3^58)

vi tich tren co thua so 13 nen tich do chia het cho 13

=

bai1

a) A=(3¹+3²)+(3³+3⁴)+...+(3⁵⁹+3⁶⁰)

=(3^1.1+3^1.3)+...+(3^59.1+3^59.2)

=3^1.(1+3)+...+3^59.(1+3)

=3^1.4+....+3^59.4

=4.(3^1+...+3^59)

vi tich tren co thua so 4 nen tich do chia het cho 4

Bài 2:(12a + 36b) = (12a + 12 x 3 x b) = 12( a + 3b)chia hết cho 12