M=75.(42013+42012+.......43+42+1)+25. Chứng minh M chia hết 100
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Ta có: A - B = 1111....1111 - 2 x 1111...111
(100 csố 1) (50 csố 1)
= 1111.....1111 x (1000...0001 - 2)
(50 chữ số 1) (có 51 csố trong đó có 49 csố 0)
= 1111.....11111 x 9999....9999
(50 csố 1) (50 csố 9)
= 1111...1111 x 9 x 1111....1111
(50csố1) (50csố1)
= (1111....1111)^2 x 3^2
= (1111.....1111 x 3)^2
Vậy hiệu A - B là một số chính phương
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A=5.415.99-4.320.89
A=5.(22)5.(32)9-22.320.(23)9
A=5.230.318-22.320.227
A=5.230.318-229.320
A=229.318.(5.2-32)
A=229.318
B=5.29.619-7.229.276
B=5.29.(2.3)9-7.229.(33)6
B=5.22.219.319-7.229.318
B=5.228.319-7.229.318
B=228.318.(5.3-7.2)
B=228.318
=>A:B= \(\frac{2^{29}.3^{18}}{2^{28}.3^{18}}\)
A:B=2
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La 5 vi o bai xet chu so tan cung cua phep tinh co noi tong 5 so tu nhien lien tiep la 5 hoac 0
VD : 1+2+3+4+5= 15
co phep tinh ma tong cua 5 so hang co tan cung la 0 nhu :
0+1+2+3+4= 10