Bài 1:

Theo đề ra ta có:

$a-2\vdots 3; a-3\vdots 5$

$a-2-2.3\vdots 3; a-3-5\vdots 5$

$\Rightarrow a-8\vdots 3; a-8\vdots 5$

$\Rightarrow a-8=BC(3,5)$

$\Rightarrow a-8\vdots 15$

$\Rightarrow a=15k+8$ với $k$ tự nhiên.

Mà $a$ chia 11 dư 6

$\Rightarrow a-6\vdots 11$

$\Rightarrow 15k+8-6\vdots 11$

$\Rightarrow 15k+2\vdots 11\Rightarrow 4k+2\vdots 11$

$\Rightarrow 4k+2-22\vdots 11\Rightarrow 4k-20\vdots 11$

$\Rightarrow 4(k-5)\vdots 11\Rightarrow k-5\vdots 11$

$\Rightarrow k=11m+5$

Vậy $a=15k+8=15(11m+5)+8=165m+83$ với $m$ tự nhiên.

Vì $a<500\Rightarrow 165m+83<500\Rightarrow m< 2,52$

$\Rightarrow m=0,1,2$

Nếu $m=0$ thì $a=165.0+83=83$

Nếu $m=1$ thì $a=165.1+83=248$

Nếu $m=2$ thì $a=165.2+83=413$

Bài 2:

$a=BC(60,85,90)$
$\Rightarrow a\vdots BCNN(60,85,90)$

$\Rightarrow a\vdots 3060$

Mà $a<1000$ nên $a=0$

a) \(A=1+2+2^2+2^3+...+2^{60}\)

=>\(2A=2+2^2+2^3+2^4+...+2^{61}\)

=>\(2A-A=\left(2+2^2+2^3+2^4+...+2^{61}\right)-\left(1+2+2^2+2^3+...+2^{60}\right)\)

=>\(A=2^{61}-1\)

b)  \(B=1+3+3^2+3^3+...+3^{46}\)

=>\(3B=3+3^2+3^3+3^4+...+3^{47}\)

=>\(3B-B=\left(3+3^2+3^3+3^4+...+3^{47}\right)-\left(1+3+3^2+3^3+...+3^{46}\right)\)

=>\(2A=3^{47}-1\)

=>\(B=\frac{3^{47}-1}{2}\)

c) \(C=1+5^2+5^4+...+5^{200}\)

=>\(5^2C=5^2+5^4+5^6+...+5^{202}\)

=>\(25C=5^2+5^4+5^6+...+5^{202}\)

=>\(25C-C=\left(5^2+5^4+5^6+...+5^{202}\right)-\left(1+5^2+5^4+...+5^{200}\right)\)

=>\(24C=5^{202}-1\)

=>\(C=\frac{5^{202}-1}{24}\)

a) A = \(1+2+2^2+2^3+...+2^{60}\)

2A = \(2.\left(1+2+2^2+2^3+...+2^{60}\right)\)

2A = \(2+2^2+2^3+2^4+...+2^{61}\)

2A - A = \(\left(2+2^2+2^3+2^4+...+2^{61}\right)\)- \(\left(1+2+2^2+2^3+...+2^{60}\right)\)

A = \(2^{61}-1\)

b)B = \(1+3+3^2+3^3+...+3^{46}\)

3B = \(3.\left(1+3+3^2+3^3+...+3^{46}\right)\)

3B = \(3+3^2+3^3+3^4+...+3^{47}\)

3B - B = \(\left(3+3^2+3^3+3^4+...+3^{47}\right)\)- \(\left(1+3+3^2+3^3+...+3^{46}\right)\)

2B = \(3^{47}-1\)

B = \(\left(3^{47}-1\right):2\)

Câu a mk ko hiểu gì nha xl bn nhìu

b)1-2+3-4+...+99-100

=(1-2)+(3-4)+...+(99-100)

=(-1)+(-1)+...+(-1)

=(-1) . 50

=(-50)

c) 5 + 5² + 5³ + ...+ 5⁹⁹ + 5¹⁰⁰ 

=(5+5²)+(5³+5⁴)+....+(5⁹⁹+5¹⁰⁰)

=30+5²(5+5²)+...+5⁹⁸(5+5²)

=30.1+5².30+.....+5⁹⁸.30

=30(1+5²+...+5⁹⁸) chia hết cho 6

Câu a:

A = 5 + 5^2 + 5^3

A = 5.(1+ 5 + 5^2)

A = 5.(1+ 5+ 25)

A = 5.(6 + 25)

A = 5.31

A ⋮ 31 (đpcm)

Câu b:

A = 5+ 5^2+ 5^3 + ..+ 5^99

Xét dãy số: 1; 2; 3; ..; 99

Dãy số trên có 99 số hạng vì 99 : 3 = 33

Nên ta nhóm 3 số hạng liên tiếp của A vào nhau khi đó:

A = (5+ 5^2+ 5^3) + ..+ (5^97+ 5^98 + 5^99)

A = 5.(1+5+5^2) + ..+ 5^97.(1+5+5^2)

A = (1+5+5^2).(5+ ..+ 5^97)

A =31.(5+..+5^97)

A ⋮ 31 (đpcm)