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4. Ҳز Ҳ

 

4.1.

x>0 x x-1 .

`. . ()=[x]+1. [x]³0 x>0, x>0 B(x)>0. [x-1]=[x]-1, ()=[x]+1>[x]=[x-1]+1=B(x-1). B(x) - .

 

4.2. :

) x<>0 x x-1 ;

) x>0 x x+1 ;

) x<=b x x+a ;

) x<>b x x+a ;

) x<y z x; x y; y z+2 .

 

4.3.

 

Circle_1

n: ;

(n);

n>1

n mod 2 =0 n n div 2

n n+1

;

(n)

.

 

4.4.

 

Circle_2

n: ;

(n);

n>1

n mod 2 =0 n n div 2+1

n n+1

;

(n)

.

 

4.5.

s 0;

n<>m

n n+1; s s+1

 

4.6. n i m :

n=q*m+r, 0<=r<m.

`. q=0, r=n.

J=(n=m*q+r)&(r>=0)

. , J , , Øa & J . , a=Ø (r<m) = (r>=m).

䳿, J a. q 䳿, , . q 1. r (q+1)*m-q*m=m, J . , q q+1; r r-m J r>=m. m ` n, . :

 

Div

m,n,q,r :;

(m,n)

(m>0)&(n>=0) ;

q 0; r n;

r>=m

q q+1; r r-m

;

(q,r)

.

 

, , r, m>0 , - B(r,m)=r-m.

 

4.7. m, n .

` 1. , , : (m,n). m=n , (m,m)=m.

m m - n m>n

n n - m n>m

n n+m; m m+n.

䳿 . B(m,n)=m+n , m<>n , B(m,n) 䳺 . , p B1(m,n) = B(m,n) - 2(m,n).

 

m,n :;

(m,n)

m>0 & n>0;

m<>n

m>n m m-- n

n n - m

;

(m)

.

, B1(m,n) .

` 2. , (m,0)=m. m n; n m mod n. B(n)=n, n<>0, B(n) 䳺 .

 

m,n ::

(m,n)

m>0 & n>0 ;

n<>0

l m; m n; n l mod n

;

(m)

.

 

4.8. , .

 

4.9. y=xn , .

`. y=zxk. z=1 k=n a y y=xn. k=0 y=zxk : y=z.

x x x, k k/2;

z z x, k k-1.

k, k. k=0.

 

n,k: ; x,z : ;

(n,x);

z 1; k n;

k <> 0

k mod 2=0

x x*x; k k div 2

z z* k; k k-1;

;

(z)

.

.

 

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