What is the solution to this equation in positive integers?

In summary, to solve equations using positive integers, follow the basic rules of algebra such as simplifying the equation and isolating the variable. There are several methods for solving these equations, including the distributive property, factoring, and substitution. Negative integers are not necessary for simple equations with positive integer solutions. To check if a solution is correct, substitute it into the original equation or use mental math. Tips for solving equations efficiently include identifying and combining like terms, using mental math, and being familiar with basic algebraic rules.
  • #1
anemone
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Solve in positive integers for

$577(bcd+b+c)=520(abcd+ab+ac+cd+1)$
 
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  • #2
anemone said:
Solve in positive integers for

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

Hello. (Sun)

Merry christmas. (Party)

[tex]577(bcd+b+c)=520a(bcd+b+c)+520(cd+1)[/tex]

[tex](577-520a)(bcd+b+c)=+520(cd+1)[/tex]

First conclusion: [tex]\ a=1[/tex]

[tex]57(bcd+b+c)=+520(cd+1)[/tex]

[tex]57b(cd+1)+57c=+520(cd+1)[/tex]

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

[tex]57 \ and \ 520 \ coprime[/tex]

[tex]57|(cd+1)[/tex]

A bit of brute force. :eek:

[tex]a=1[/tex]
[tex]b=9[/tex]
[tex]c=7[/tex]
[tex]d=8[/tex]

Regards.
 
  • #3
mente oscura said:
Hello. (Sun)

Merry christmas. (Party)

[tex]577(bcd+b+c)=520a(bcd+b+c)+520(cd+1)[/tex]

[tex](577-520a)(bcd+b+c)=+520(cd+1)[/tex]

First conclusion: [tex]\ a=1[/tex]

[tex]57(bcd+b+c)=+520(cd+1)[/tex]

[tex]57b(cd+1)+57c=+520(cd+1)[/tex]

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

[tex]57 \ and \ 520 \ coprime[/tex]

[tex]57|(cd+1)[/tex]

A bit of brute force. :eek:

[tex]a=1[/tex]
[tex]b=9[/tex]
[tex]c=7[/tex]
[tex]d=8[/tex]

Regards.

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

so b = 9
 
  • #4
kaliprasad said:
57c=(520−57b)(cd+1)
=> 520 - 57b < 57
or b >= 9

so b = 9

[tex]57c=7(cd+1)[/tex]

[tex]c(57-7d)=7 \ \rightarrow{d=8} \ \rightarrow{c=7}[/tex]

Regards.
 
  • #5
mente oscura said:
Hello. (Sun)

Merry christmas. (Party)

[tex]577(bcd+b+c)=520a(bcd+b+c)+520(cd+1)[/tex]

[tex](577-520a)(bcd+b+c)=+520(cd+1)[/tex]

First conclusion: [tex]\ a=1[/tex]

[tex]57(bcd+b+c)=+520(cd+1)[/tex]

[tex]57b(cd+1)+57c=+520(cd+1)[/tex]

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

[tex]57 \ and \ 520 \ coprime[/tex]

[tex]57|(cd+1)[/tex]

A bit of brute force. :eek:

[tex]a=1[/tex]
[tex]b=9[/tex]
[tex]c=7[/tex]
[tex]d=8[/tex]

Regards.

mente oscura said:
[tex]57c=7(cd+1)[/tex]

[tex]c(57-7d)=7 \ \rightarrow{d=8} \ \rightarrow{c=7}[/tex]

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!
 

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

1. How do you solve equations using positive integers?

To solve equations using positive integers, you need to follow the basic rules of algebra. Start by simplifying the equation and combining like terms. Then, isolate the variable on one side of the equation by performing inverse operations. Finally, solve for the variable by dividing both sides of the equation by the coefficient. If the answer is a decimal, round up to the nearest whole number to get the solution in positive integers.

2. Is there a specific method for solving equations with positive integers?

Yes, there are several methods for solving equations with positive integers. Some common methods include using the distributive property, factoring, and the substitution method. It is important to choose the method that best suits the given equation and simplifies the solution process.

3. Can negative integers be used in solving equations with positive integers?

No, the purpose of solving equations with positive integers is to find solutions that are only positive whole numbers. Negative integers can be used in more complex equations, but for simple equations with positive integer solutions, negative integers are not necessary.

4. How can you check if your solution to an equation with positive integers is correct?

To check if your solution is correct, you can substitute the value of the variable into the original equation and see if it satisfies the equation. If it does, then your solution is correct. Another way is to use mental math and perform the inverse operations in your head to see if it leads to the original equation.

5. Are there any tips for solving equations with positive integers efficiently?

Yes, some tips for solving equations with positive integers efficiently include identifying and combining like terms, using mental math to simplify calculations, and being familiar with basic algebraic rules. It is also helpful to double-check your solution and make sure it satisfies the original equation.

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