a)vì 60.n chia hết cho 15

     45 chia hết cho 15

=>60.n+45 chia hết cho 15

b)vì 60.n chia hết cho 30

       45 ko chia hết cho 30

    =>60.n +45 ko chia hết cho 30

Ta có : 36³⁶ - 9¹⁰ = (4.9)³⁶ - 9¹⁰ = 4³⁶ . 9³⁶ - 9¹⁰ = 9¹⁰(4³⁶ . 9²⁶ -1) \(⋮\)9   (1)

            36³⁶ có chữ số tận cùng là 6 và 9¹⁰ có chữ số tận cùng là 1 (vì số mũ là số lẻ)

            => 36³⁶ - 9¹⁰ có chữ số tận cùng là 5   

            => 36³⁶ - 9¹⁰ \(⋮\)5    (2)

Từ (1) và (2) suy ra : 36³⁶ - 9¹⁰ \(⋮\)5;9      (3)

Mà 5 và 9 là hai số nguyên tố cùng nhau   (4)

Từ (3) và (4) suy ra : 36³⁶ - 9¹⁰ \(⋮\)45 

Vậy 36³⁶ - 9¹⁰ \(⋮\)45 (điều cần phải chứng minh)

1223344567890654564255

Áp dụng hằng đẳng thức sau
an−1=(a−1).[an−1+an−2+...+1]=(a−1).pan−1=(a−1).[an−1+an−2+...+1]=(a−1).p (nn là 1 số nguyên dương)
an+1=(a+1).[an−1−an−2+..+1]=(a+1).qan+1=(a+1).[an−1−an−2+..+1]=(a+1).q (nn là 1 số nguyên dương lẻ)

Thay vào ta được như sau:

+) 222333−1=(222−1).p=13.17.p222333−1=(222−1).p=13.17.p

+) 333222+1=(3332)111+1=110889111+1=(110889+1).q=13.8530.q333222+1=(3332)111+1=110889111+1=(110889+1).q=13.8530.q

=>=> 222333+333222=222333−1+333222+1=13(17p+8530q)⋮13222333+333222=222333−1+333222+1=13(17p+8530q)⋮13

Vậy: 222333+333222⋮13222333+333222⋮13 (đpcm)(đpcm) 

a)\(10^9+10^8+10^7=10^7\left(10^2+10+1\right)=10^7\cdot111=2\cdot10^6\cdot555⋮555\)

b)\(81^7-27^9-9^{13}=3^{28}-3^{27}-3^{26}=3^{26}\left(3^2-3-1\right)=3^{26}.5=3^{24}\cdot45⋮45\)

Chúc bạn học tốt :)!

\(\left(222^{333}+333^{222}\right)⋮13\)

an−1=(a−1).[an−1+an−2+...+1]=(a−1).p" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">an−1=(a−1).[an−1+an−2+...+1]=(a−1).pn" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">nan+1=(a+1).[an−1−an−2+..+1]=(a+1).q" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">an+1=(a+1).[an−1−an−2+..+1]=(a+1).qn" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">n222333−1=(222−1).p=13.17.p" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">222333−1=(222−1).p=13.17.p
333222+1=(3332)111+1=110889111+1=(110889+1).q=13.8530.q" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">333222+1=(3332)111+1=110889111+1=(110889+1).q=13.8530.q
=>" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">=>222333+333222=222333−1+333222+1=13(17p+8530q)⋮13" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">\(222\)333+333222=222333−1+333222+1=13(17p+8530q)⋮13

a) \(222^{333}+333^{222}\)

\(=\left(111.2\right)^{333}+\left(111.3\right)^{222}\)

\(=111^{333}.2^{333}+111^{222}.3^{222}\)

\(=111^{222}.\left(111^{111}.2^{333}+3^{222}\right)\)

\(=111^{222}.\left(111^{111}.2^{3.111}+3^{2.111}\right)\)

\(=111^{222}.\left[111^{111}.\left(2^3\right)^{111}+\left(3^2\right)^{111}\right]\)

\(=111^{222}.\left(111^{111}.8^{111}+9^{111}\right)\)

\(=111^{222}.\left[\left(111.8\right)^{111}+9^{111}\right]\)

\(=111^{222}.\left(888^{111}+9^{111}\right)\)

\(=111^{222}.\left(888+9\right)\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]\)

\(=111^{222}.7992\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]\)

\(=111^{222}.897\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]\)

\(=111^{222}.13.69\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]⋮13\)

Vậy \(222^{333}+333^{222}⋮13\left(dpcm\right)\)

Ta có 7¹⁰⁰⁰=(7⁴)²⁵⁰=(...1)²⁵⁰=(...1)

         3¹⁰⁰⁰=(3⁴)²⁵⁰=(...1)²⁵⁰=(...1)

         =>7¹⁰⁰⁰-3¹⁰⁰⁰=(...1)-(...1)=(...0)=>chia hết cho 10=> điều phải cm

Chúc bn học tốt!!

#Zon_của_Dôn