MHB What is the solution to this equation in positive integers?

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The equation $577(bcd+b+c)=520(abcd+ab+ac+cd+1)$ is being discussed for solutions in positive integers. Participants confirm that a solution provided by one user is correct and aligns with another's findings. The conversation includes holiday greetings, indicating a friendly atmosphere among participants. The focus remains on solving the mathematical problem, with acknowledgment of correct answers. The discussion highlights collaboration and validation of solutions in the context of integer equations.
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Solve in positive integers for

$577(bcd+b+c)=520(abcd+ab+ac+cd+1)$
 
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anemone said:
Solve in positive integers for

$577(bcd+b+c)=520(abcd+ab+ac+cd+1)$

Hello. (Sun)

Merry christmas. (Party)

577(bcd+b+c)=520a(bcd+b+c)+520(cd+1)

(577-520a)(bcd+b+c)=+520(cd+1)

First conclusion: \ a=1

57(bcd+b+c)=+520(cd+1)

57b(cd+1)+57c=+520(cd+1)

57c=(520-57b)(cd+1) \ \rightarrow{b<10}

57 \ and \ 520 \ coprime

57|(cd+1)

A bit of brute force. :o

a=1
b=9
c=7
d=8

Regards.
 
mente oscura said:
Hello. (Sun)

Merry christmas. (Party)

577(bcd+b+c)=520a(bcd+b+c)+520(cd+1)

(577-520a)(bcd+b+c)=+520(cd+1)

First conclusion: \ a=1

57(bcd+b+c)=+520(cd+1)

57b(cd+1)+57c=+520(cd+1)

57c=(520-57b)(cd+1) \ \rightarrow{b<10}

57 \ and \ 520 \ coprime

57|(cd+1)

A bit of brute force. :o

a=1
b=9
c=7
d=8

Regards.

57c=(520−57b)(cd+1)
=> 520 - 57b < 57
or b >= 9

so b = 9
 
kaliprasad said:
57c=(520−57b)(cd+1)
=> 520 - 57b < 57
or b >= 9

so b = 9

57c=7(cd+1)

c(57-7d)=7 \ \rightarrow{d=8} \ \rightarrow{c=7}

Regards.
 
mente oscura said:
Hello. (Sun)

Merry christmas. (Party)

577(bcd+b+c)=520a(bcd+b+c)+520(cd+1)

(577-520a)(bcd+b+c)=+520(cd+1)

First conclusion: \ a=1

57(bcd+b+c)=+520(cd+1)

57b(cd+1)+57c=+520(cd+1)

57c=(520-57b)(cd+1) \ \rightarrow{b&lt;10}

57 \ and \ 520 \ coprime

57|(cd+1)

A bit of brute force. :o

a=1
b=9
c=7
d=8

Regards.

mente oscura said:
57c=7(cd+1)

c(57-7d)=7 \ \rightarrow{d=8} \ \rightarrow{c=7}

Regards.

Bravo, mente oscura! Your answer is correct and your solution is pretty much the same as the one that I have and you're simply awesome!

Merry Christmas to you too!

kaliprasad said:
57c=(520−57b)(cd+1)
=> 520 - 57b < 57
or b >= 9

so b = 9

Yes, that's correct, kaliprasad and thanks for participating!
 

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