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 #1
avatar+18625 
+2

Express a4(b - c) + b4 (c - a) + c4 (a - b) as the product of four factors.

(b-c),(c-a) and (a-b) are products of this expression

 

\(\begin{array}{|rcll|} \hline a^4(b-c)+b^4(c-a)+c^4(a-b) &=& (b-c)(c-a)(a-b)\cdot x \\\\ x &=& \dfrac{a^4(b-c)+b^4(c-a)+c^4(a-b)}{(b-c)(c-a)(a-b)} \\\\ &=& \dfrac{a^4b-a^4c+b^4c-b^4a+c^4a-c^4b}{-a^2b+a^2c-b^2c+b^2a-c^2a+c^2b} \\\\ &=& \dfrac{ a^4·b - a^4·c - a·b^4 + a·c^4 + b^4·c - b·c^4 }{-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2} \\\\ \hline \end{array}\)

 

Polynom division after variable c:

\(\small{ \begin{array}{|rcll|} \hline && \text {in divisor term with max power of c is: } -ac^2 \\ && \text {current residue: } a^4·b - a^4·c - a·b^4 + a·c^4 + b^4·c - b·c^4 \\ && \text {in current residue term with max power of c: } ac^4 \\ && \text {quotient } \frac{ac^4}{-ac^2} = -c^2 \\ && \text {product } -c^2·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^2·b·c^2 - a^2·c^3 - a·b^2·c^2 + a·c^4 + b^2·c^3 - b·c^4 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^4·b - a^4·c - a^2·b·c^2 + a^2·c^3 - a·b^4 + a·b^2·c^2 + b^4·c - b^2·c^3 \\ && \text {in current residue term with next lower power of c is: } a^2·c^3 \\ && \text {quotient } \frac{a^2·c^3}{-ac^2} = -a·c \\ && \text {product } -a·c·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^3·b·c - a^3·c^2 - a^2·b^2·c + a^2·c^3 + a·b^2·c^2 - a·b·c^3 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^4·b - a^4·c - a^3·b·c + a^3·c^2 + a^2·b^2·c - a^2·b·c^2 - a·b^4 + a·b·c^3 + b^4·c - b^2·c^3 \\ && \text {in current residue term with next lower power of c is: } a·b·c^3 \\ && \text {quotient } \frac{a·b·c^3}{-a·c^2} = -b·c \\ && \text {product } -b·c·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^2·b^2·c - a^2·b·c^2 - a·b^3·c + a·b·c^3 + b^3·c^2 - b^2·c^3 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^4·b - a^4·c - a^3·b·c + a^3·c^2 - a·b^4 + a·b^3·c + b^4·c - b^3·c^2 \\ && \text {in current residue term with next lower power of c is: } a^3·c^2 \\ && \text {quotient } \frac{a^3·c^2}{-a·c^2} = -a^2 \\ && \text {product } -a^2·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^4·b - a^4·c - a^3·b^2 + a^3·c^2 + a^2·b^2·c - a^2·b·c^2 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^3·b^2 - a^3·b·c - a^2·b^2·c + a^2·b·c^2 - a·b^4 + a·b^3·c + b^4·c - b^3·c^2 \\ && \text {in current residue term with next lower power of c is: } a^2·b·c^2 \\ && \text {quotient } \frac{a^2·b·c^2}{-a·c^2} = -a·b \\ && \text {product } -a·b·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^3·b^2 - a^3·b·c - a^2·b^3 + a^2·b·c^2 + a·b^3·c - a·b^2·c^2 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^2·b^3 - a^2·b^2·c - a·b^4 + a·b^2·c^2 + b^4·c - b^3·c^2 \\ && \text {in current residue term with next lower power of c is: } a·b^2·c^2 \\ && \text {quotient } \frac{a·b^2·c^2}{-a·c^2} = -b^2 \\ && \text {product } -b^2·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^2·b^3 - a^2·b^2·c - a·b^4 + a·b^2·c^2 + b^4·c - b^3·c^2 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } 0 \\\\ x&=& \dfrac{a^4·b - a^4·c - a·b^4 + a·c^4 + b^4·c - b·c^4 } {-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2 } \\\\ &=& -a^2 - a·b - a·c - b^2 - b·c - c^2 \\ \hline \end{array} }\)

 

\( a^4(b-c)+b^4(c-a)+c^4(a-b) = (b-c)(c-a)(a-b)(-a^2 - a·b - a·c - b^2 - b·c - c^2) \)

 

laugh

heureka 20.10.2017 12:02
 #1
avatar+18625 
+1

What is the smallest positive integer that will satisfy the following two modular equations:

N mod 3,331 =1,851 and

N mod 1,851 =1,468?

 

\(\begin{array}{|rcll|} \hline N &\equiv& 1851 \pmod{3331} \\ N &\equiv& 1468 \pmod{1851} \\ \hline \end{array}\)

 

Formula:

\(\begin{array}{rcll} N = 1851\cdot 1851\cdot \frac{1}{1851} \pmod{3331} + 1468\cdot 3331 \cdot \frac{1}{3331} \pmod{1851} + 3331\cdot 1851\cdot z \\ \quad z \in Z \\ \end{array}\)

 

check:

\(\begin{array}{|rcll|} \hline \pmod{3331}: & N = 1851 \cdot \frac{1851}{1851} + 0 +0 = 1851 \ \checkmark \\ \pmod{1851}: & N = 0 + 1468 \cdot \frac{3331}{3331} + 0 = 1468 \ \checkmark \\ \hline \end{array}\)

 

\(\text{with }\varphi() =\text{ Euler's totient theorem }: \\ 3331 \text{ prime number} \qquad \varphi(p) = p-1 \qquad \varphi(3331) = 3330 \\ 1851 =3\cdot 617\qquad \varphi(1851) =1851\cdot \left(1-\frac13\right)\left(1-\frac{1}{617}\right) = 1232 \)

 

Modular inverses:

\(\begin{array}{|rcll|} \hline && \frac{1}{1851} \pmod{3331} \\ &\equiv& 1851^{\varphi(3331)-1} \pmod{3331} \quad & | \quad gcd(3331,1851)=1 \\ &\equiv& 1851^{3330-1} \pmod{3331} \\ &\equiv& 1851^{3329} \pmod{3331} \\ && \ldots \\ &\equiv& 835 \pmod{3331} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline && \frac{1}{3331} \pmod{1851} \\ &\equiv& 3331^{\varphi(1851)-1} \pmod{1851} \quad & | \quad gcd(3331,1851)=1 \\ &\equiv& 3331^{1232-1} \pmod{1851} \\ &\equiv& 3331^{1231} \pmod{1851} \\ && \ldots \\ &\equiv& 1387 \pmod{1851} \\ \hline \end{array}\)

 

\(\begin{array}{rcll} N &=& 1851\cdot 1851\cdot 835 + 1468 \cdot 3331 \cdot 1387 + 3331\cdot 1851\cdot z \quad & | \quad z \in Z \\ &=& 2860877835 + 6782302396 + 6165681\cdot z \\ &=& \underbrace{9643180231}_{\equiv 55147 \pmod{6165681}} + 6165681\cdot z \\ &=& 55147 + 6165681\cdot z \\ \end{array} \)

 

\(\begin{array}{|rcll|} \hline \mathbf{55147} &\mathbf{\equiv}& \mathbf{1851 \pmod{3331}} \\ \mathbf{55147} &\mathbf{\equiv}& \mathbf{1468 \pmod{1851}} \\ \hline \end{array}\)

 

laugh

heureka 20.10.2017
 #1
avatar+18625 
+1
heureka 16.10.2017