Question

CMR: B= 3+3^3+3^5+...+3^1991 chia hết cho 13 và 41

Answer 1

De thay B co 996 so hang

Ta co: 3+3^3+3^5+...+3^1991

= (3+3^3+3^5)+...+(3^1987+1989+1991)

=3.(1+3^2+3^4)+...+3^1987.(1+3^2+3^4)

=3.91+...+3^1987.91

=(3+..+3^1987).91=(3+...+3^1987).13.7 chia het cho 13

 

3+3^3+3^5+...+3^1991

=(3+3^3+3^5+3^7)+...+(3^1985+3^1987+3^1989+3^1991)

=3(1+3^2+3^4+3^6)+...+3^1985.(1+3^2+3^4+3^6)

=3.820+...+3^1985.820=(3+...+3^1985).820=(3+....+3^1985).41.20 chia het cho 41

Answer 2

chưng tỏ B:13

B=3+3³+3⁵+...+3¹⁹⁹¹:13

B=3. (1+9+81)+3⁷.(1+9+81)+...+3¹⁹⁸⁹.(1+9+81):13

B=91.(3+3⁷+3¹³+...+3¹⁹⁸⁹):13

vì 91:13=>B:13

vậy B:13

chưng tỏ B:41

B=3+3³+3⁵+...+3¹⁹⁹¹:41

B=3.(1+9+81+729)+3⁹.(1+9+81+729)+...+3¹⁹⁸⁸.(1+9+81+729):41

B=820.(3+3⁹+3¹⁷+...+3¹⁹⁸⁸):41

vì 820:41=>B:41

vậy B:41

Answer 3

B=3+3³+3⁵+...+3¹⁹⁹¹:13

B=3. (1+9+81)+3⁷.(1+9+81)+...+3¹⁹⁸⁹.(1+9+81):13

B=91.(3+3⁷+3¹³+...+3¹⁹⁸⁹):13

vì 91:13=>B:13

vậy B:13

chưng tỏ B:41

B=3+3³+3⁵+...+3¹⁹⁹¹:41

B=3.(1+9+81+729)+3⁹.(1+9+81+729)+...+3¹⁹⁸⁸.(1+9+81+729):41

B=820.(3+3⁹+3¹⁷+...+3¹⁹⁸⁸):41

vì 820:41=>B:41

vậy B:41