. , : + = .
, , .
1. , ≥ .
. , .. , ≥ , . ≥ , ($ k Î N 0)[ a = + k ]. , k Î N 0.
. .., , ≥ . , k, .. = k. , :
+ ( ) = + k, = + k Þ ≥ .
2. , .
. $ c 1, c 2 Î N 0, , = 1 = 2. = + 1 = + 2, .. + 1 = + 2 Þ 1 = 2 , .., 1> 2 Þ + 1> + 2 ( 8).
3. (" a, , Î N 0) [ a ≥ Þ ( ) = ].
.
. = [ + ( )] = + ( ) , ..
= + ( ) , ( ) = . .
. : , = .
, , .
. , , 0.
= 0, ≠ 0. : = , = . 0, 0. .. ≠ 0, = 0. , , .
≠ 0, = 0. = , , 0, 0. , .
= = 0. = . .
. , , ≤ . , .
.
, .. , = . , 1 ≤ . , ≤ . = , ≤ .
|
|
.
, q 1 q 2, q 1 = , q 2 = q 1 = q 2 Þ (q 1 q 2) = 0. .. ≠ 0, q 1 q 2 = 0, , q 1 = q 2.
1. . , ,
( + ): = : + : .
. q 1 , q 2 , .. = q 1, = q 2, , + = cq 1 + cq 2 = c (q 1 + q 2). , + , q 1 + q 2, .. : + : .
: : ( + ) ≠ : + : .
, 66: (2 + 4) ≠ 66: 2 + 66: 4.
2. . , ≥ , ( ): = : : .
. ( ).
. 192: 4 = (200 8): 4 = 200: 4 8: 4 = 50 2 = 48.
3. . , ( ): = (: ) . , ( ): = (: ) .
. , = , ,
= = ( ) ( ): = = (: ) = (: ) .
. 560: 7 = (56 10): 7 = (56: 7) 10 = 80.
4. . : ( ) = (: ): =
= (: ): . , : ( ) = (: ): . : ( ) = t Þ = ( ) t. : = (( ) t): = t. (: ): = (c t): = t.
. 480: 60 = 480: (6 10) = (480: 10): 6 = 48: 6 = 8.
, . : . 48 4 , 4 48. 12. , 48: 4 = 12.
:
7368 24
72 307
168
3 :
73: 24 = 3 (. 1);
16: 24 = 0 (. 16);
168: 24 = 7 (. 0).
: 11:2 = 5 (. 1), 19: 4 = 4 (. 3). .
. , q r , :
1) = q + r,
2) 0 ≤ r < .
. q r .
.
, , : = q (.0), , = q + 0, 0 ≤ 0 < .
, 0 < < , : = 0 (. ), , = 0 + , 0 ≤ < .
, > (.86: 10), . , :
|
|
1, 2, 3,..., q, (q + 1), .
, .. ≥ 1. , , , .. . q, q < a (q + 1) > . a > q, q , .. q = r, r Î N 0, .. (q + 1) > , q + > > q, .. > r. , r < . , :
1) = q + r,
2) 0 ≤ r < .
(, ) (q, r).
.
, (, ) 2 : 2 2 , :
.
, , r 1 < , r 2 < , r 2 r 1 < ,
r 2 r 1 = 0. , r 1 = r 2 q 1 = q 2. .
. , , , , .
, : = q (. r), :
1) = q + r,
2) 0 ≤ r < .
1) 2) Î N, :
1) am = q + r,
2) 0 ≤ rm < m.
1. ?
2. , , {1, 2, 3,... } .
3. ..., n
12 + 32 + 52 +... + (2 n 1)2 = .
4. , : ) 3 + 2; ) 3 + 3; ) 3 + 4.
5. , : ) 3 2; ) 3 3; ) 3 4.
6. , , 30 8; 30 6; 30 31.
.
VII