tìm số dư trong phép chia ( theo Ptp đồng dư) a)1^5+3^5+5^5+....+75^5+99^5:4 b) 35^2 -35^3+35^4-35^8+35^16+35^32:425
Tìm số dư trong phép chia (theo đồng dư) \(35^2-35^3+35^4-35^8+35^{16}+35^{32}\) cho \(425\).
Các bạn giúp tôi với ạ. :((
Ta có: \(35^2\equiv375\)( mod 425)
\(35^3=35.35^2\equiv35.375\equiv375\)( mod 425)
\(35^4=35.35^3\equiv35.375\equiv375\)( mod 425)
\(35^8=35^4.35^4\equiv375.375\equiv375\)( mod 425)
\(35^{16}\equiv35^8.35^8\equiv375.375\equiv375\)( mod 425)
\(35^{32}\equiv35^{16}.35^{16}\equiv375.375\equiv375\)( mod 425)
=> \(35^2-35^3+35^4-35^8+35^{16}+35^{32}\equiv375-375+375-375+375+375\equiv325\)( mod 425)
Vậy số dư cần tìm là 325
1_ Tìm số tự nhiên nhỏ nhất chia cho 5 dư 3, chia 7 dư 4 và chia 9 dư 5.
2_ Tìm số tự nhiên <500 sao cho chia nó cho 15 dư 8, chia nó cho 35 dư 13
3_ Tìm số tự nhiên nhỏ nhất chia cho 3 và 5 cùng được số dư là 1, chia cho 4 dư 3
3: \(\left\{{}\begin{matrix}a-1\in\left\{15;30;45;...\right\}\\a-3\in\left\{4;8;12;...\right\}\end{matrix}\right.\Leftrightarrow a=31\)
A = 9/8 - 8/9 + 3/25 + 1/4 - 5/16 + 19/25 - 1/9 + 2/25 - 1/81
B = -1/3- 8/35 - 2/9 - 1/35 + 4//5 + -4/9 + 3/7
\(A=\dfrac{9}{8}-\dfrac{8}{9}+\dfrac{3}{24}+\dfrac{1}{4}-\dfrac{5}{16}+\dfrac{19}{25}-\dfrac{1}{9}+\dfrac{2}{25}-\dfrac{1}{81}\)
\(=\dfrac{9}{8}+\dfrac{1}{4}-\dfrac{5}{16}+\dfrac{1}{8}-\dfrac{8}{9}-\dfrac{1}{9}-\dfrac{1}{81}+\dfrac{19}{25}+\dfrac{2}{25}\)
\(=\dfrac{10}{8}+\dfrac{1}{4}-\dfrac{5}{16}-1-\dfrac{1}{81}+\dfrac{21}{25}\)
\(=\dfrac{20+4-5}{16}-\dfrac{82}{81}+\dfrac{21}{25}\)
\(=\dfrac{19}{16}-\dfrac{82}{81}+\dfrac{21}{25}\)
\(=\dfrac{32891}{16\cdot81\cdot25}\)
b: \(B=-\dfrac{1}{3}-\dfrac{8}{35}-\dfrac{2}{9}-\dfrac{1}{35}+\dfrac{4}{5}-\dfrac{4}{9}+\dfrac{3}{7}\)
\(=\dfrac{-1}{3}-\dfrac{2}{9}-\dfrac{4}{9}-\dfrac{8}{35}-\dfrac{1}{35}+\dfrac{4}{5}+\dfrac{3}{7}\)
\(=\dfrac{-3-2-4}{9}+\dfrac{-9}{35}+\dfrac{28+15}{35}\)
\(=-1+\dfrac{-9+43}{35}=-1+\dfrac{34}{35}=-\dfrac{1}{35}\)
a. ( 19 5/8 : 7/12 - 13 1/4 : 7/12) . 4/5
b.( - 2/5 + 3/7) - (4/9 + 12/20 - 13/35) + 7/35
c. ( 4 5/57 - 3/ 4/51 + 8 13/29) - ( 3 5/57 - 6 16/29)
\(\left(19\frac{5}{8}:\frac{7}{12}-13\frac{1}{4}:\frac{7}{12}\right).\frac{4}{5}\)
\(=\left(\frac{157}{8}:\frac{7}{12}-\frac{53}{4}:\frac{7}{12}\right).\frac{4}{5}\)
\(=\left[\left(\frac{157}{8}-\frac{53}{4}\right):\frac{7}{12}\right].\frac{4}{5}\)
\(=\left[\frac{51}{8}:\frac{7}{12}\right].\frac{4}{5}\)
\(=\frac{153}{14}.\frac{4}{5}\)
\(=\frac{306}{35}\)
\(\left(\frac{-2}{5}+\frac{3}{7}\right)-\left(\frac{4}{9}+\frac{12}{20}-\frac{13}{35}\right)+\frac{7}{35}\)
\(=\frac{1}{35}-\frac{212}{315}+\frac{7}{35}\)
\(=\frac{1}{35}+\frac{-212}{315}+\frac{7}{35}\)
\(=\frac{9}{315}+\frac{-212}{315}+\frac{63}{315}\)
\(=\frac{-140}{315}=\frac{-4}{9}\)
\(\left(4\frac{5}{57}-3\frac{4}{51}+8\frac{13}{29}\right)-\left(3\frac{5}{57}-6\frac{16}{29}\right)\)
\(=\left(\frac{233}{57}-\frac{157}{51}+\frac{245}{29}\right)-\left(\frac{176}{57}-\frac{190}{29}\right)\)
\(=\frac{1941}{323}-\frac{-5726}{1653}\)
\(=\frac{14011}{1479}\)
1/2×(2×x-3)+105/2=-137/2 A,-12×25-25×75-25×13 B,-50/7×49/10-35/2×-10/7+ -25/3× -9/5 C,-354+265-156+125 D,-74+40-50+16-35 E,-2/3×4/5+-4/5×4/3-0,125
1/2×(2×x-3)+105/2=-137/2
1/2×(2×x-3)=-137/2-105/2
1/2×(2×x-3)=-242/2
2×x-3=-242/2:1/2
2×x-3=-242
2.x=(-242)+3
2.x=239
x=239:2
x=239/2
A,-12×25-25×75-25×13
=[(-12)-75-13].25
=(-100).25
-2500
B,-50/7×49/10-35/2×-10/7+ -25/3× -9/5
=(-35)-(-25)+(-13)
=-23
C,-354+265-156+125
=[(-354)-156]+(265+125)
=(-510)+277
=-233
D,-74+40-50+16-35
=(-74)+40+(-50)+16+(-35)
=[(-74)+(-35)]+[40+(-50)+16]
=(-109)+26
=-83
E,-2/3×4/5+-4/5×4/3-0,125
=-2/3.4/5+4/5.(-4/3)-0,125
=4/5.[-2/3+(-4/3)]-0,125
=4/5.(-2)-0,125
=-8/5-0,125
=(-1,6)+(-0,125)
=-1,725
3.Quy đồng mẫu các phân số sau
A.7/10 ; -5/-15 và 3/17 B.-4/-75 ; -3/5 và 8/35
\(a,MSC:170\\ \dfrac{7}{10}=\dfrac{7.17}{17.10}=\dfrac{119}{170}\\ \dfrac{-5}{15}=-\dfrac{1}{5}=\dfrac{1.34}{5.34}=\dfrac{34}{170}\\ \dfrac{3}{17}=\dfrac{3.10}{17.10}=\dfrac{30}{17}\\ b,MSC:525\\ \dfrac{-4}{-75}=\dfrac{4}{75}=\dfrac{4.7}{75.7}=\dfrac{28}{525}\\ \dfrac{-3}{5}=\dfrac{-3.105}{5.105}=\dfrac{-315}{525}\\ \dfrac{8}{35}=\dfrac{8.15}{35.15}=\dfrac{120}{525}\)
Tìm số dư trong phép chia \(35^2-35^3+35^4-35^8+35^{16}+35^{32}\) cho 425
cho số tự nhiên A lớn hơn 35 biết A chia 5 dư 1,A chia 7 dư 5,A chia 35 dư 6
thế 100 chia 100 trừ 100 chia 100000000 bằng bao nhiêu
chứng minh :
A= 1+3+3^2+3^3+3^4+...+3^11 , chia hết cho 52
B= 7+7^3+7^5+...+7^2017, chia hết cho 35
S=1+5+5^2+5^3+5^4+...+5^2010
Tìm số dư khi chia S cho 2, cho 20, cho 13