Với a,b \(\ne\)0. Chứng minh rằng:
a. \(\overline{abba}\) \(⋮\)11
b. \(\overline{aaabbb}\)\(⋮\)37
c. \(\overline{ababab}\)\(⋮\)7
d. \(\overline{abab}\)- \(\overline{baba}\)\(⋮\)9, 11 (a > b)
chứng minh rằng a) \(\overline{abcabc}\) chia hết cho 7, 11, 13
b) \(\overline{ab}-\overline{ba}\) chia hết cho 9
c) \(\overline{abc}-\overline{cba}\) chia hết cho 99
a) Ta có: \(\overline{abcabc}=100000a+10000b+1000c+100a+10b+c\) \(=100100a+10010b+1001c\) \(=1001\left(100a+10b+c\right)=7\cdot11\cdot13\left(100a+10b+c\right)⋮7,11,13\)
b) Ta có: \(\overline{ab}-\overline{ba}=10a+b-10b-a=9a-9b\) \(=9\left(a-b\right)⋮9\)
c) Ta có: \(\overline{abc}-\overline{cba}=100a+10b+c-100c-10b-a=99a-99c=99\left(a-c\right)⋮99\)
Bài 1:Chứng minh rằng
a) \(\overline{ab}\) = 2.\(\overline{cd}\) → \(\overline{abcd}\) ⋮ 67
b) Cho \(\overline{abc⋮27}\) chứng minh rằng \(\overline{bca}\) ⋮ 27
Bài 2: Chứng minh rằng: Nếu \(\overline{ab}\) + \(\overline{cd}\) ⋮11 thì \(\overline{abcd}\) ⋮11
Bài 1:
a)
\(\overline{abcd}=100\overline{ab}+\overline{cd}\)
\(=100.2\overline{cd}+\overline{cd}\)
\(=201\overline{cd}\)
Mà \(201⋮67\)
\(\Rightarrow\overline{abcd}⋮67\)
b)
\(\overline{abc}=100\overline{a}+10\overline{b}+\overline{c}\)
\(=\left(100\overline{b}+10\overline{c}+\overline{a}\right)+\left(99\overline{a}-90\overline{b}-9\overline{c}\right)\)
\(=\overline{bca}+9\left[\left(12\overline{a}-9\overline{b}\right)-\left(\overline{a}+\overline{b}+\overline{c}\right)\right]\)
\(=\overline{bca}+27\left(4\overline{a}-3\overline{b}\right)-\left(\overline{a}+\overline{b}+\overline{c}\right)⋮27\)
\(\Rightarrow\overline{bca}-\left(\overline{a}+\overline{b}+\overline{c}\right)⋮27\)
\(\Rightarrow\left\{{}\begin{matrix}\overline{bca}⋮27\\\overline{a}+\overline{b}+\overline{c}⋮27\end{matrix}\right.\)
\(\Rightarrow\overline{bca}⋮27\)
Bài 2:
\(\overline{abcd}=\overline{ab}.100+\overline{cd}\)
\(=\overline{ab}.99+\overline{ab}+\overline{cd}\)
\(=\overline{ab}.11.99+\left(\overline{ab}+\overline{cd}\right)\)
Mà \(11⋮11\)
\(\Rightarrow\overline{ab}.11.9⋮11\)
\(\Rightarrow\overline{abcd}⋮11\).
Các bạn giải nhanh cho mình nhé. Thanks!
Hai phân số sau có bằng nhau không? \(\frac{\overline{abab}}{\overline{cdcd}};\frac{\overline{ababab}}{\overline{cdcdcd}}\)
Hai phân số sau có bằng nhau không?
=\(\frac{ab}{cd}\)
\(\Rightarrow\) Hai phan so đều bằng nhau.
bằng nhau đó nha
abab/cdcd=101/101.ab/cd =ab/cd
ababab/cdcdcd=10101/10101.ab/cd=ab/cd
vì ab/cd=ab/cd
suy ra abab/cdcd=ababab/cdcdcd
Chứng tỏ rằng:
\(\overline{ab}\) - \(\overline{ba}\) \(⋮\) 9 ( a > b )
\(\overline{abba}\) \(⋮\) 11
ab - ba ⋮ 9
ab - ba=a * 10+b*1-b*10-a*1
=a*(10-1)-b*(10-1)=a*9-b*9=9*(a-b)⋮9(vì 9⋮9)
vậy ab-ba⋮9
abba ⋮ 11
abba=a*1000+b*100+b*10+a.1=a*(1000+1)+b*(100+10)
=a*1001+b*110=a*11*91+b*10*11=11(a*91+b*10)⋮11(vì 11⋮11)
Vậy abba⋮11
ab - ba ⋮ 9
ab - ba=a x 10+b x 1-b x 10-a x 1
=a x (10-1)-b x (10-1)=a x 9-b x 9=9x (a-b)⋮9(vì 9⋮9)vậy ab-ba⋮9abba ⋮11
abba=a x 1000+b x 100+b x 10+a.1= a x (1000+1)+b x (100+10)
=a x 1001+b x 110=a x 11 x 91+b x 10 x 11=11(a x 91+b x 10)⋮11(vì 11⋮11)Vậy abba⋮11
ab - ba ⋮ 9
ab - ba=a * 10+b*1-b*10-a*1
=a*(10-1)-b*(10-1)=a*9-b*9=9*(a-b)⋮9(vì 9⋮9)
vậy ab-ba⋮9
abba ⋮ 11
abba=a*1000+b*100+b*10+a.1=a*(1000+1)+b*(100+10)
=a*1001+b*110=a*11*91+b*10*11=11(a*91+b*10)⋮11(vì 11⋮11)
Vậy abba⋮11
chứng minh rằng: a) nếu \(\overline{ab}+\overline{cd}+\overline{eg}\) \(⋮\) 11 thì \(\overline{abcdeg}\) \(⋮\) 11
Ta có: \(\overline{abcdeg}=10000\overline{ab}+100\overline{cd}+\overline{eg}=9999\overline{ab}+99\overline{cd}+\left(\overline{ab}+\overline{cd}+\overline{eg}\right)⋮11\)
Tìm các số tự nhiên có dạng \(\overline{abba}\)thỏa mãn điều kiện :
\(\overline{abba=}\overline{ab^2+}\overline{ba^2+a}-b\)
Cho dãy tỉ số \(\frac{\overline{ab}+\overline{bc}}{a+b}=\frac{\overline{bc}+\overline{ca}}{b+c}=\frac{\overline{ca}+\overline{ab}}{c+a}\)( với a,b,c\(\ne\)0 ) .Tính \(P=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có :
\(\frac{\overline{ab}+\overline{bc}}{a+b}=\frac{\overline{bc}+\overline{ca}}{b+c}=\frac{\overline{ca}+\overline{ab}}{c+a}=\frac{\overline{ab}+\overline{bc}+\overline{bc}+\overline{ca}+\overline{ca}+\overline{ab}}{a+b+b+c+c+a}=\frac{2\left(\overline{ab}+\overline{bc}+\overline{ca}\right)}{2\left(a+b+c\right)}=\frac{\overline{ab}+\overline{bc}+\overline{ca}}{a+b+c}\)
\(=\frac{10a+b+10b+c+10c+a}{a+b+c}=\frac{11a+11b+11c}{a+b+c}=\frac{11\left(a+b+c\right)}{a+b+c}=11\)
Lại có : \(P=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}\)
+) Nếu \(a+b+c=0\) :
\(\Rightarrow\)\(a+b=-c\)
\(\Rightarrow\)\(b+c=-a\)
\(\Rightarrow\)\(a+c=-b\)
Thay \(a+b=-c\)\(;\)\(b+c=-a\) và \(a+c=-b\) vào \(\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}\) ta được :
\(\frac{-c}{b}.\frac{-a}{c}.\frac{-b}{a}=\frac{-\left(abc\right)}{abc}=-1\)
+) Nếu \(a+b+c\ne0\) :
Do đó :
\(\frac{\overline{ab}+\overline{bc}}{a+b}=11\)\(\Rightarrow\)\(10a+11b+c=11a+11b\)\(\Rightarrow\)\(c=a\)\(\left(1\right)\)
\(\frac{\overline{bc}+\overline{ca}}{b+c}=11\)\(\Rightarrow\)\(10b+11c+a=11b+11c\)\(\Rightarrow\)\(a=b\)\(\left(2\right)\)
\(\frac{\overline{ca}+\overline{ab}}{c+a}=11\)\(\Rightarrow\)\(10c+11a+b=11c+11a\)\(\Rightarrow\)\(b=c\)\(\left(3\right)\)
Từ (1), (2) và (3) suy ra :
\(a=b=c\)
Suy ra :
\(P=\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}=\frac{b+b}{b}.\frac{c+c}{c}.\frac{a+a}{a}=\frac{2b}{b}.\frac{2c}{c}.\frac{2a}{a}=2.2.2=8\)
Vậy \(P=-1\) hoặc \(P=8\)
Chúc bạn học tốt ~
hai phân số sau có bằng nhau không?
\(\frac{\overline{abab}}{\overline{cdcd}}\); \(\frac{ababab}{\overline{cdcdcd}}\)
Cho \(\dfrac{a+\overline{bc}}{\overline{abc}}=\dfrac{b+\overline{ca}}{\overline{bca}}=\dfrac{c+\overline{ab}}{\overline{cab}}\). Chứng minh rằng \(\dfrac{\overline{ab}}{c}=\dfrac{\overline{ca}}{b}=\dfrac{\overline{bc}}{a}\)