Ghi ch\u00fa: T\u00e0i li\u1ec7u tr\u00edch d\u1eabn<\/p>\n\n\n\n
C\u00f9ng Vietlearn \u00f4n b\u00e0i: Trong b\u00e0i h\u1ecdc tr\u01b0\u1edbc, c\u00e1c b\u00e9 \u0111\u00e3 \u0111\u01b0\u1ee3c t\u00ecm hi\u1ec3u c\u00e1ch ph\u00e2n t\u00edch m\u1ed9t s\u1ed1 ra th\u1eeba s\u1ed1 nguy\u00ean t\u1ed1. T\u1ea1i b\u00e0i h\u1ecdc h\u00f4m nay, Vietlearn s\u1ebd ti\u1ebfp t\u1ee5c gi\u1edbi thi\u1ec7u v\u1edbi c\u00e1c con ki\u1ebfn th\u1ee9c v\u1ec1 \u01b0\u1edbc chung v\u00e0 b\u1ed9i chung. H\u00e3y c\u00f9ng b\u1eaft \u0111\u1ea7u b\u00e0i h\u1ecdc.<\/p>\n\n\n\n
\u01af\u1edbc chung v\u00e0 b\u1ed9i chung<\/p>\n\n\n\n
L\u01b0u \u00fd: Ta ch\u1ec9 x\u00e9t \u01b0\u1edbc chung v\u00e0 b\u1ed9i chung c\u1ee7a s\u1ed1 kh\u00e1c 0.<\/p>\n\n\n\n
S\u1edf d\u0129 kh\u00f4ng x\u00e9t \u01b0\u1edbc chung v\u00e0 b\u1ed9i chung c\u1ee7a s\u1ed1 0 l\u00e0 v\u00ec 0 kh\u00f4ng chia \u0111\u01b0\u1ee3c cho ch\u00ednh n\u00f3 v\u00e0 s\u1ed1 0 c\u0169ng kh\u00f4ng c\u00f3 b\u1ed9i.<\/p>\n\n\n\n
\u01af\u1edbc chung l\u00e0 g\u00ec<\/p>\n\n\n\n
V\u00ed d\u1ee5: Vi\u1ebft t\u1eadp h\u1ee3p c\u00e1c \u01b0\u1edbc c\u1ee7a 8 v\u00e0 t\u1eadp h\u1ee3p c\u00e1c \u01b0\u1edbc c\u1ee7a 10, ta c\u00f3:<\/p>\n\n\n\n
\u01af\u208d\u2084\u208e = {1; 2; 4, 8}<\/p>\n\n\n\n
\u01af\u208d\u2086\u208e = {1; 2; 5; 10}<\/p>\n\n\n\n
So s\u00e1nh c\u00e1c ph\u1ea7n t\u1eed trong hai t\u1eadp h\u1ee3p ta th\u1ea5y s\u1ed1 1 v\u00e0 2 v\u1eeba l\u00e0 \u01b0\u1edbc c\u1ee7a 8 v\u1eeba l\u00e0 \u01b0\u1edbc c\u1ee7a 10. Ta n\u00f3i ch\u00fang l\u00e0 c\u00e1c \u01b0\u1edbc chung c\u1ee7a 8 v\u00e0 10.<\/p>\n\n\n\n
\u01af\u1edbc chung c\u1ee7a hai hay nhi\u1ec1u s\u1ed1 l\u00e0 \u01b0\u1edbc c\u1ee7a t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 \u0111\u00f3.<\/p>\n\n\n\n
Ta k\u00ed hi\u1ec7u t\u1eadp h\u1ee3p c\u00e1c \u01b0\u1edbc chung c\u1ee7a 8 v\u00e0 10 l\u00e0 \u01afC(8, 10). Ta c\u00f3:<\/p>\n\n\n\n
\u01afC(8, 10) = {1; 2}.<\/p>\n\n\n\n
x \u2208 \u01afC\u208d\u0430,b\u208e n\u1ebfu a \u22ee x v\u00e0 b \u1367 x<\/p>\n\n\n\n
T\u01b0\u01a1ng t\u1ef1 ta c\u0169ng c\u00f3:<\/p>\n\n\n\n
x \u2208 \u01afC\u208d\u0430,b,c\u208e n\u1ebfu a \u22ee x, b \u1367 x v\u00e0 c \u1367 x<\/p>\n\n\n\n
C\u00e2u h\u1ecfi:<\/p>\n\n\n\n
Kh\u1eb3ng \u0111\u1ecbnh sau \u0111\u00e2y \u0111\u00fang hay sai:<\/p>\n\n\n\n
8 \u2208 \u01afC\u208d16,40\u208e; 8 \u2208 \u01afC\u208d32,28\u208e<\/p>\n\n\n\n
L\u1eddi gi\u1ea3i:<\/p>\n\n\n\n
Kh\u1eb3ng \u0111\u1ecbnh 8 \u2208 \u01afC\u208d16,40\u208e \u0111\u00fang v\u00ec 16 \u22ee 8 v\u00e0 40 \u22ee 8<\/p>\n\n\n\n
Kh\u1eb3ng \u0111\u1ecbnh 8 \u2208 \u01afC\u208d32,28\u208e sai v\u00ec 32 \u1367 8 nh\u01b0ng 28 kh\u00f4ng chia h\u1ebft cho 8<\/p>\n\n\n\n
B\u1ed9i chung l\u00e0 g\u00ec<\/p>\n\n\n\n
V\u00ed d\u1ee5:<\/p>\n\n\n\n
Vi\u1ebft t\u1eadp h\u1ee3p A c\u00e1c b\u1ed9i c\u1ee7a 3 v\u00e0 t\u1eadp h\u1ee3p B c\u00e1c b\u1ed9i c\u1ee7a 5, ta c\u00f3:<\/p>\n\n\n\n
A = {0; 3; 6; 9; 12; 15; 18; 21; 24; 27; 30\u2026}<\/p>\n\n\n\n
B = {0; 5; 10; 15; 20; 25; 30\u2026}<\/p>\n\n\n\n
X\u00e9t t\u1eadp h\u1ee3p A v\u00e0 B, ta th\u1ea5y c\u00e1c s\u1ed1 0; 15; 30,\u2026 l\u00e0 b\u1ed9i c\u1ee7a 3 v\u00e0 5. Ta n\u00f3i ch\u00fang l\u00e0 b\u1ed9i chung c\u1ee7a 3 v\u00e0 5.<\/p>\n\n\n\n
T\u01b0\u01a1ng t\u1ef1 nh\u01b0 \u01b0\u1edbc chung, ta c\u00f3 k\u1ebft lu\u1eadn sau:<\/p>\n\n\n\n
B\u1ed9i chung c\u1ee7a hai hay nhi\u1ec1u s\u1ed1 l\u00e0 b\u1ed9i c\u1ee7a t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 \u0111\u00f3.<\/p>\n\n\n\n
Ta k\u00ed hi\u1ec7u t\u1eadp h\u1ee3p c\u00e1c b\u1ed9i chung c\u1ee7a 3 v\u00e0 5 l\u00e0 BC(3, 9).<\/p>\n\n\n\n
x \u2208 BC\u208d\u0430,b\u208e n\u1ebfu x \u22ee a v\u00e0 x \u1367 b<\/p>\n\n\n\n
T\u01b0\u01a1ng t\u1ef1 ta c\u0169ng c\u00f3:<\/p>\n\n\n\n
x \u2208 BC\u208d\u0430,b,c\u208e n\u1ebfu x \u22ee a, x \u1367 b v\u00e0 x \u1367 c<\/p>\n\n\n\n
C\u00e2u h\u1ecfi: \u0110i\u1ec1n s\u1ed1 v\u00e0o \u00f4 vu\u00f4ng \u0111\u1ec3 \u0111\u01b0\u1ee3c m\u1ed9t kh\u1eb3ng \u0111\u1ecbnh \u0111\u00fang:<\/p>\n\n\n\n
6 \u2208 BC\u208d3,\ud83b\ude84\u208e<\/p>\n\n\n\n
L\u1eddi gi\u1ea3i:<\/p>\n\n\n\n
6 \u2208 BC\u208d3, 2\u208e<\/p>\n\n\n\n
6 \u2208 BC\u208d3, 1\u208e<\/p>\n\n\n\n
Ta c\u00f3 th\u1ec3 \u0111i\u1ec1n s\u1ed1 2 ho\u1eb7c s\u1ed1 1 v\u00e0 \u00f4 tr\u1ed1ng v\u00ec 6 chia h\u1ebft cho 2 v\u00e0 1<\/p>\n\n\n\n
T\u1eadp h\u1ee3p \u01afC\u208d4,6\u208e = {1; 2}, bao g\u1ed3m c\u00e1c ph\u1ea7n t\u1eed chung c\u1ee7a 2 t\u1eadp h\u1ee3p \u01af\u208d4\u208e v\u00e0 \u01af\u208d6\u208e. T\u1eadp \u01afC(4, 6) \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 l\u00e0 giao c\u1ee7a 2 t\u1eadp h\u1ee3p \u01af\u208d4\u208e v\u00e0 \u01af\u208d6\u208e.<\/p>\n\n\n\n
Giao c\u1ee7a hai t\u1eadp h\u1ee3p (t\u1eadp con) l\u00e0 m\u1ed9t t\u1eadp h\u1ee3p g\u1ed3m c\u00e1c ph\u1ea7n t\u1eed chung c\u1ee7a hai t\u1eadp h\u1ee3p \u0111\u00f3.<\/p>\n\n\n\n
Ta k\u00fd hi\u1ec7u giao c\u1ee7a t\u1eadp h\u1ee3p A v\u00e0 B l\u00e0 A \u2312 B.<\/p>\n\n\n\n
Nh\u01b0 v\u1eady \u01af\u208d4\u208e \u2312 \u01af\u208d6\u208e = \u01afC\u208d4, 6\u208e ; B\u208d4\u208e \u2312 B\u208d6\u208e = BC\u208d4, 6\u208e.<\/p>\n\n\n\n
V\u00ed d\u1ee5:<\/p>\n\n\n\n
A = {3, 4, 6}<\/p>\n\n\n\n
B = {4, 6}<\/p>\n\n\n\n
Ta th\u1ea5y gi\u1eefa hai t\u1eadp h\u1ee3p A v\u00e0 B c\u00f3 c\u00e1c ph\u1ea7n t\u1eed chung l\u00e0 4 v\u00e0 6. V\u00ec v\u1eady, ta \u0111\u01b0\u1ee3c t\u1eadp giao gi\u1eefa A v\u00e0 B l\u00e0:<\/p>\n\n\n\n
A \u2312 B = {4, 6}<\/p>\n\n\n\n
X = {a, b}<\/p>\n\n\n\n
Y = {c}<\/p>\n\n\n\n
Ta th\u1ea5y gi\u1eefa hai t\u1eadp h\u1ee3p X v\u00e0 Y kh\u00f4ng ch\u1ee9a b\u1ea5t k\u1ef3 ph\u1ea7n t\u1eed chung n\u00e0o. V\u00ec v\u1eady, t\u1eadp giao gi\u1eefa X v\u00e0 Y l\u00e0 t\u1eadp r\u1ed7ng.<\/p>\n\n\n\n
X \u2312 Y = \u0424<\/p>\n\n\n\n
M\u1eb9o ghi nh\u1edb:<\/p>\n\n\n\n
N\u1ebfu c\u00e1c s\u1ed1 a,b,c \u0111\u1ec1u chia h\u1ebft cho x th\u00ec a, b, c l\u00e0 b\u1ed9i chung c\u1ee7a x v\u00e0 x \u0111\u1ed3ng th\u1eddi l\u00e0 \u01b0\u1edbc chung c\u1ee7a a, b,c.<\/p>\n\n\n\n
\u01af\u1edbc chung ph\u1ea3i l\u00e0 m\u1ed9t s\u1ed1 kh\u00e1c 0<\/p>\n\n\n\n
M\u1ecdi s\u1ed1 \u0111\u1ec1u c\u00f3 \u01b0\u1edbc chung l\u00e0 1<\/p>\n\n\n\n
Do ch\u1ec9 c\u00f3 duy nh\u1ea5t 2 \u01b0\u1edbc, s\u1ed1 nguy\u00ean t\u1ed1 ch\u1ec9 c\u00f3 m\u1ed9t \u01b0\u1edbc chung l\u00e0 1<\/p>\n\n\n\n
M\u1ecdi s\u1ed1 kh\u00e1c 0 \u0111\u1ec1u c\u00f3 b\u1ed9i chung l\u00e0 0<\/p>\n\n\n\n
L\u00fd gi\u1ea3i: S\u1ed1 kh\u00f4ng chia h\u1ebft cho m\u1ecdi s\u1ed1 kh\u00e1c ch\u00ednh n\u00f3 v\u00e0 m\u1ecdi s\u1ed1 \u0111\u1ec1u chia h\u1ebft cho 1<\/p>\n\n\n\n
T\u1ea3i ngay: B\u1ed9 \u0111\u1ec1 thi v\u00e0 b\u00e0i h\u1ec7 th\u1ed1ng ki\u1ebfn th\u1ee9c m\u00f4n To\u00e1n tuy\u1ec7t hay<\/p><\/blockquote>\n\n\n\n
- B\u00e0i t\u1eadp v\u1ec1 \u01b0\u1edbc chung v\u00e0 b\u1ed9i chung<\/li><\/ol>\n\n\n\n
B\u00e0i t\u1eadp 134:<\/p>\n\n\n\n
B\u00e0i t\u1eadp 134 s\u00e1ch gi\u00e1o khoa<\/p>\n\n\n\n
L\u1eddi gi\u1ea3i:<\/p>\n\n\n\n
a, 4 \u2209 \u01afC(12, 18)<\/p>\n\n\n\n
V\u00ec 12 chia h\u1ebft cho 4 nh\u01b0ng 18 kh\u00f4ng chia h\u1ebft cho 4<\/p>\n\n\n\n
b, 6 \u2208 \u01afC(12,18)<\/p>\n\n\n\n
V\u00ec 12 chia h\u1ebft cho 6 v\u00e0 18 c\u0169ng chia h\u1ebft cho 6<\/p>\n\n\n\n
c, 2 \u2208 \u01afC(4, 6, 8)<\/p>\n\n\n\n
V\u00ec 4, 6, 8 \u0111\u1ec1u chia h\u1ebft cho 2<\/p>\n\n\n\n
d, 4 \u2209 \u01afC(4, 6, 8)<\/p>\n\n\n\n
V\u00ec 4 v\u00e0 8 chia h\u1ebft cho 4 nh\u01b0ng 6 kh\u00f4ng chia h\u1ebft cho 4<\/p>\n\n\n\n
e, 80 \u2209 BC(20, 30)<\/p>\n\n\n\n
V\u00ec 80 chia h\u1ebft cho 20 nh\u01b0ng kh\u00f4ng chia h\u1ebft cho 30<\/p>\n\n\n\n
g, 60 \u2208 BC(20, 30)<\/p>\n\n\n\n
V\u00ec 60 chia h\u1ebft cho c\u1ea3 20 v\u00e0 30<\/p>\n\n\n\n
h, 12\u2209 BC( 4, 6, 8)<\/p>\n\n\n\n
V\u00ec 12 chia h\u1ebft cho 4 v\u00e0 6 nh\u01b0ng kh\u00f4ng chia h\u1ebft cho 8<\/p>\n\n\n\n
i, 24 \u2208 BC(4, 6, 8)<\/p>\n\n\n\n
V\u00ec 24 chia h\u1ebft cho 4, 6 v\u00e0 8<\/p>\n\n\n\n
B\u00e0i t\u1eadp 135:<\/p>\n\n\n\n
B\u00e0i t\u1eadp 135 s\u00e1ch gi\u00e1o khoa<\/p>\n\n\n\n
L\u1eddi gi\u1ea3i:<\/p>\n\n\n\n
a,<\/p>\n\n\n\n
\u01af(6) = {1, 2, 3, 6}<\/p>\n\n\n\n
\u01af(9) = {1, 3, 9}<\/p>\n\n\n\n
Ta th\u1ea5y gi\u1eefa hai t\u1eadp h\u1ee3p tr\u00ean c\u00f3 c\u00e1c ph\u1ea7n t\u1eed chung l\u00e0 1 v\u00e0 3. T\u1eeb \u0111\u00f3, ta c\u00f3:<\/p>\n\n\n\n
\u01afC(6, 9) = {1, 3}<\/p>\n\n\n\n
b,<\/p>\n\n\n\n
\u01af(7) = {1, 7}<\/p>\n\n\n\n
\u01af(8) = {1, 2, 4, 8}<\/p>\n\n\n\n
Ta th\u1ea5y gi\u1eefa hai t\u1eadp h\u1ee3p tr\u00ean ch\u1ec9 c\u00f3 ph\u1ea7n t\u1eed chung l\u00e0 1. T\u1eeb \u0111\u00f3, ta c\u00f3:<\/p>\n\n\n\n
\u01afC(7, 8) = {1}<\/p>\n\n\n\n
c.<\/p>\n\n\n\n
\u01af(4) = {1, 2, 4}<\/p>\n\n\n\n
\u01af(6) = {1, 2, 3, 6}<\/p>\n\n\n\n
\u01af(8) = {1, 2 , 4 , 8)<\/p>\n\n\n\n
Ta th\u1ea5y gi\u1eefa hai t\u1eadp h\u1ee3p tr\u00ean c\u00f3 ph\u1ea7n t\u1eed chung l\u00e0 1 v\u00e0 2. T\u1eeb \u0111\u00f3, ta c\u00f3:<\/p>\n\n\n\n
\u01afC(4, 6, 8) = {1, 2}<\/p>\n\n\n\n
B\u00e0i t\u1eadp 136:<\/p>\n\n\n\n
B\u00e0i t\u1eadp 136 s\u00e1ch gi\u00e1o khoa<\/p>\n\n\n\n
L\u1eddi gi\u1ea3i:<\/p>\n\n\n\n
T\u1eadp h\u1ee3p c\u00e1c s\u1ed1 t\u1ef1 nhi\u00ean l\u00e0 b\u1ed9i c\u1ee7a 6 v\u00e0 nh\u1ecf h\u01a1n 40:<\/p>\n\n\n\n
A ={0, 6, 12, 18, 24, 30, 36}<\/p>\n\n\n\n
T\u1eadp h\u1ee3p c\u00e1c s\u1ed1 t\u1ef1 nhi\u00ean l\u00e0 b\u1ed9i c\u1ee7a 9 v\u00e0 nh\u1ecf h\u01a1n 40:<\/p>\n\n\n\n
B = {0, 9, 18, 27, 36}<\/p>\n\n\n\n
a, T\u1eadp h\u1ee3p c\u00e1c ph\u1ea7n t\u1eed c\u1ee7a M l\u00e0:<\/p>\n\n\n\n
M = {0, 18, 36}<\/p>\n\n\n\n
b, M\u1ed1i quan h\u1ec7 gi\u1eefa M v\u1edbi A v\u00e0 B:<\/p>\n\n\n\n
M \u2282 A<\/p>\n\n\n\n
M \u2282 B<\/p>\n\n\n\n