, 䳿.
( ) . , . ,
453 4 = (4 102 + 5 10 + 3) 4 = (4 102) 4 + (5 10) 4 + 3 4.
, :
(4 4) 102 + (5 4) 10 + (3 4) = 16 102 + 20 10 + 12 = 1812.
, , , . 10 k k . 10. 10 , . ,
453 132 = 453 (1 102 + 3 10 + 2) = (453 1) 102 + (453 3) 10 + (453 2).
, :
× 132
+1359
453___
= an an-1 a1 a0 = bm bm-1 b1b0 :
1. .
2. b0 b0 .
3. b1, .
4. bm.
5. .
ij , . () . . :
1) , , , 63 9;
2) , 65 9: 65 = 9 7 + 2, 65: 9 = 7 (. 2).
, 䒺 , 䒺 q r, = bq + r, 0 ≤ r < b.
bq ≤ a < b (q + 1), q, . r = b q. , 637 25 䒺 q r, 637 = 25 ∙ q + r.
25 q ≤ 637 < 25(q +1) q. , 25 ∙ 10 < 637 < 25 100, q . 25 10, 20,... 25 ∙ 20 < 637 <25 ∙ 30, 2, 20 < q < 30, q = 20 + q1, q1 . 25 ∙ (20 + q 1) ≤ 637 < 25 ∙ (20 + q 1 +1),
|
|
500 + 25 q1 ≤ 637 < 500 + 25(q1 +1), 25 q1 ≤ 137 < 25(q1 +1).
q1 . , 25 1, 2, 3,... ij: 25 5 = 125, 25 ∙ 6 = 150. 5. , q = 25, a r = 637 635 = 12
637 = 25 25 + 12. :
_ 637 25
50 25
_ 137
125
䒺 b :
1. = b, q = 1, r = 0.
2. > b b , , b 1, 2,...,9, q b r = bq.
3. > b , b, .
, b , 1, , b; 1, 2,..., 9 q1, bq1 ≤ c1, bq1 c1 .
ij r1 = 1 bq1. bq1; r1 b; b, , b, , b, .