- k 1 k 2. X k 1 - .
(2) - , (1). - . k 2 (2); i k 1 k 2 - - , (2) - k 2.
1. X(2) = 1011,1001 -
- . , k 1< k 2 , - (, -). - , (2):
, k 1> k 2, - . .
k 1 - k 2(k 1> k 2) . k 2 k 1 . , k 2. - , k 2 . , k 2 , , , . . .
2. (10) = 1247 .
, k 1< k 2, , ( ) k 1, - ( ) . k 1 . , ( ). k 2- .
3. (2) = 10110110.
k 1- k 2 (k 1> k 2 ) . , , k 2 k 1 . , - , k 2 . - , . (. . ) , . .
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4. (10) = 0,314 , .
k 1< k 2 , - , k 1 . 1 , , - . k 2- .
5. (2) = 0,101101.
k 2 . , - , k 2 . k 2- , - , , - . - , - , .
6. (10) = 1247 .
(. 2) {5) = 14442. - {-2, -1, 0, 1, 2}, 4 - : 4 = . . 1247, .
- .
7. .
- .
k 1- - {1, } {0, 1). , , 00...01, . . 1 . - 00...01 1...11, 1 .
8. X (10)= 0,314 (1, }. 4, :
X (2)= 0,0101000001; = 0,1 1 1 .
, .
2 r , . 2 r - 2 r r .
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9. X (8) = 762,15 Y (16) = 51,7D - .
X (2) = 111110010,001101; Y (2) =1110010100011010,01111101.
2 r , , , - no r , . r - 2 r.
10. {2) = 11111101, 1000001 -
- . - k 1 k 2, . . . k 2( }. k 1 1, , k 1+ 1+1 .
k < 1
xi i - . ,
k 1 k 2< -1. - k 2 . B. i ( i = 0) i, i .
i , i i +1- |k| - xi. A , i +, i . (n -) 1 | k | 1 + 2- + 1- .
, i i+ 1- |k| xi i - . - . i , - i i - . .
11. X (10) = 30,75 Y (10) = 23,5 - k 2= 2. 2 X (2) = 11110,11 Y (2)= 10111,1.
1 1 1 1 0, 1 1 1 0 1 1 1, 1
, X (-2) = 1100011,11; Y (-2) = 111001,1.
k 2 , B. , . , . . , , , , . .
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12. X (-2) = 10100,1101 Y (-2) = 101101,10101. , 24, - 25.
k - . - , k. - D, , -
- X i (i = ). - X . - .
- , :
rest (A+B) (mod Pi) = (rest A (mod Pi) + rest B (mod Pi)) (mod Pi),
rest AB (mod Pi) = (rest A (mod Pi) x rest B (mod Pi)) (mod Pi). (4)
i (i = ) k -
k j = (k 1 j, k 2 j, , k m j), (j = ),
k -
xj = (x 1 j , x 2 j , , x m j ), (j = ).
Xi = rest X (mod Pi) = rest (mod Pi) =
= (rest xj (mod Pi) rest k j (mod Pi)) (mod Pi) = (xi j k i j (mod Pi)) (mod Pi).
, X Pi, - xj k j. Pi.
, - (4), :
rest AB (mod Pi) = (A rest B (mod Pi)) (mod Pi).
i - kj, j, ..
Xi = (x j k i j (mod Pi)) (mod Pi) = (xi j k j (mod Pi)) (mod Pi). (5)
13. X (10)= 839 1= 3, 2 = 5, 3 = 7, P 4=11. -
102= (1, 0, 2, 1); 8 = (2, 3, 1, 8);
101=(1, 0, 3, 10); 3 = (0, 3, 3, 3);
100= (1, 1, 1, 1); 9 = (0, 4, 2, 9).
Xi:
1= (1ּ2 +1ּ0 +1ּ0) (mod 3) = 2;
X 2 = (0ּ3 + 0ּ3 + 1ּ 4) (mod 5) = 4;
3= (2ּ1 + 3ּ3 +1ּ2) (mod 7) = 6;
4= (1ּ8 +10ּ3 +1ּ9) (mod 11) = 3.
, X = (2, 4, 6, 3). - (5).
- ,
X = l 1 P 1 + X 1; X = l 2 P 2 + X 2, ; X = li Pi + Xi,
, X, l 1, l 2 , , lm , , - . , .
( ) 1 , 2 , , m , Pi (i = )
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1 = (1, 0, 0, , 0);
2 = (0, 1, 0, , 0);
...;
m = (0, 0, 0, , 1).
X k - - :
. i k - Bi(k)
Bi(k) = .
i - ,
rest (mod Pi) = 1.
14. X = (2, 3, 4, 5), 1= 3, 2 = 5, 3 = 7, 4 = 11. N = 3ּ5ּ7ּ11 = 1155; = 1, = 1, = 2, = 2; 1= 385, B 2= 231, B 3 = 330, B 4 = 210. ,
X (10) = 2ּ385 + 3ּ231 + 4ּ330 + 5ּ210 (mod 1155) = 3833 (mod 1155) = 368.
- , . - Pi (i = ) qi i- qi -1 Pi,a q o = 1. X
,
bi . b 0= 1, - bi
bi = rest ai (mod Pi +1),
,
.
bi -1, , i 1; Pi j; (i ≠ j). , Pi,
rest Pi (mod j) = 1.
15. 1= 5, 2 = 9, 3= 11 X = (3, 6, 10). bi - ij .
rest 5 P 12 (mod 9) = 1; rest 9 P 21 (mod 5) = 1;
rest 5 P 13 (mod 11) = 1; rest 9 P 23 (mod 11) = 1
12 = 2, P 13= 9; 21 = 6, 23= 5. , b 0= 3, = (3, 6, 10); = (3, 3, 3); = (0, 2, 9).
, 1(0) = ((3, 6, 10) - (3, 3, 3)) (0, 2, 9) = (0, 3, 7) (0, 2, 9) = (0,6, 8),
b 1 = rest a 1 (mod 9) = 6;
2(0) = ((0, 6, 8) - (0, 6, 6)) (6, 0, 5) = (0, 0, 2) (6, 0, 5) = (0,0, 10),
b 2 = rest a 2 (mod 11) = 10.
, X (10) = b 0 + b 1 P 1 + b 2 P 1 P 2 = 3 + 6ּ5 + 10ּ5ּ9 = 483.