anonymous
  • anonymous
Suppose a,b, and c are positive integers such that a+b+c+ab+bc+ca+abc=1000. Find a+b+c.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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asnaseer
  • asnaseer
You have posted the same question again. same hint applies - try to factor the expression on the left
anonymous
  • anonymous
i dont know how to do that.
asnaseer
  • asnaseer
and make an attempt to factor it - you will soon see a pattern evolve

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asnaseer
  • asnaseer
*at least make an attempt
asnaseer
  • asnaseer
I can start you off...
asnaseer
  • asnaseer
a + b + c + ab + bc + ca + abc = 1000 a(1+b) + b + c + bc + ca + abc = 1000 a(1+b) + c(1+b) + b + ca + abc = 1000 a(1+b) + c(1+b) + ca(1+b) + b = 1000 a(1+b) + c(1+b) + ca(1+b) + b + 1 - 1 = 1000 a(1+b) + c(1+b) + ca(1+b) + (1+b) = 1001 can you continue from here?
anonymous
  • anonymous
let me try
anonymous
  • anonymous
no idea
asnaseer
  • asnaseer
can you there is a common factor of (1+b) in all the terms on the left of the equals sign?
anonymous
  • anonymous
yes
asnaseer
  • asnaseer
so first pull that common factor out - what do you end up with?
asnaseer
  • asnaseer
a(1+b) + c(1+b) + ca(1+b) + (1+b) = (1+b)( ... ) = 1001
anonymous
  • anonymous
(1+b)( a+c+ac+1) = 1001
anonymous
  • anonymous
then?
asnaseer
  • asnaseer
good - now notice the two terms that have a common factor of "c" - factor those next
anonymous
  • anonymous
ok
asnaseer
  • asnaseer
what do you end up with?
anonymous
  • anonymous
(1+a)(1+b)(1+c)=1001
asnaseer
  • asnaseer
perfect!
asnaseer
  • asnaseer
now notice the right hand side - split it into a product of prime numbers
anonymous
  • anonymous
how?
asnaseer
  • asnaseer
find the factors of the number 1001
asnaseer
  • asnaseer
hint - try dividing it by 11
anonymous
  • anonymous
91*11
asnaseer
  • asnaseer
now try factoring 91
anonymous
  • anonymous
7*13
asnaseer
  • asnaseer
perfect!
asnaseer
  • asnaseer
so you ended up with: (1+a)(1+b)(1+c)=1001=7*11*13
asnaseer
  • asnaseer
since the product on the right involves primes numbers only, you can assert that each term on the left is equal to one of these primes
anonymous
  • anonymous
oh i see thank you sir thank you very much
asnaseer
  • asnaseer
yw :)

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