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\(1;a,942^{60}-351^{37}\)
\(=\left(942^4\right)^{15}-\left(....1\right)\)
\(=\left(....6\right)^{15}-\left(...1\right)\)
\(=\left(...6\right)-\left(...1\right)=\left(....5\right)⋮5\)
\(b,99^5-98^4+97^3-96^2\)
\(=\left(...9\right)-\left(...6\right)+\left(...3\right)-\left(...6\right)\)
\(=\left(...6\right)-\left(...6\right)=\left(...0\right)⋮2;5\)
\(2;5n-n=4n⋮4\)
B= 1+ 5+ 5^2+ 5^3+ ... + 5^96+ 5^97+ 5^98
=(1+5+52)+(53+54+55)+....+(596+597+598)
=31+(53.1+53.5+53.52)+....+(596.1+597.5+598.52)
=31+53.(1+5+52)+....+596.(1+5+52)
=31.1+53.31+...+596.31
=31.(1+53+...+596)
=> B chia hết cho 31
B = 1+5+52+53+....+598
B = (1+5+52)+(53+54+55)+....+(596+597+598)
B = 1(1+5+52)+53(1+5+52)+....+596(1+5+52)
B = 1.31 + 53.31+.......+596.31
B = 31.(1+53+.....+596) chia hết cho 31 (đpcm)
gom: (1+5+5^2)+(5^3+5^4+5^5)+....(5^402+5^403+5^404)
=1(1+5+5^2)+5^3(1+5+5^2)+...+5^402(1+5+5^2)
=1.31+5^3.31+...+5^402.31
Vay 1+5+5^2+...+5^403+5^404chia het cho 31
\(S=5+5^2+5^3+5^4+...+5^{2022}\\ =\left(5+5^2\right)+5^2.\left(5+5^2\right)+...+5^{2020}.\left(5+5^2\right)\\ =30+30.5^2+...+30.5^{2020}\\ =30.\left(1+5^2+...+5^{2020}\right)⋮30\)
\(S=5+5^2+5^3+...+5^{2022}\)
\(\Rightarrow S=\left(5+5^2\right)+5^2\left(5+5^2\right)+...+5^{2000}\left(5+5^2\right)\)
\(\Rightarrow S=20+5^2.20+...+5^{2000}.20\)
\(\Rightarrow S=20\left(1+5^2+...+5^{2000}\right)⋮20\)
\(\Rightarrow dpcm\)
1+5+5^2+...+5^99=(1+5+5^2)+5^3x(1+5+5^2)+5^6x(1+5+5^2)+...+5^97x(1+5+5^2) [vì có 99 số hạng chia hết cho 3]
=31+5^3x31+5^6x31+...+5^97x31=(1+5^3+5^6+...+5^97)x31 chia hết cho 31
B=1+5+52+53+...+596+597+598
=(1+5+52)+(53+54+55)+...+(596+597+598)
=31+53.(1+5+52)+...+596.(1+5+52)
=31+53.31+...+596.31
=31.(1+53+...+596)
=>B chia hết cho 31
B = 5 + 5² + 5³ + ... + 5⁹⁰
= (5 + 5² + 5³) + (5⁴ + 5⁵ + 5⁶) + ... + (5⁸⁸ + 5⁸⁹ + 5⁹⁰)
= 5.(1 + 5 + 5²) + 5⁴.(1 + 5 + 5²) + ... + 5⁸⁸.(1 + 5 + 5²)
= 5.31 + 5⁴.31 + ... + 5⁸⁸.31
= 31.(5 + 5⁴ + ...+ 5⁸⁸) ⋮ 31
Vậy B ⋮ 31
\(B=5+5^2+5^3+...+5^{89}+5^{90}\)
Ta có: \(B=\left(5+5^2+5^3\right)+...+\left(5^{88}+5^{89}+5^{90}\right)\)
\(B=155+...+5^{87}.\left(5+5^2+5^3\right)\)
\(B=155+...+5^{87}.155\)
\(B=155.\left(1+...+5^{87}\right)\)
Vì \(155⋮31\) nên \(155.\left(1+...+5^{87}\right)⋮31\)
Vậy \(B⋮31\)
\(#WendyDang\)