Question

There exist positive integers A, B and C with no common factors greater than 1, such that Alog 5 + Blog 2 = C.
The sum A+B+C equals (Here the base of the log is 200)
A+B+C = 7
A+B+C = 6
A-B+C=0
A+B-C=0​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{A, B and C are positive intgers with no common factors}$

$\mathsf{greater\;than\;1\;and\;A\,log5+B\,log2=C}$

$\underline{\textbf{To find:}}$

$\textsf{The value of A+B+C}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{A\,log\,5+B\,log\,2=C}$

$\textsf{This can be written as,}$

$\mathsf{log\,5^A+log\,2^B=C}$

$\mathsf{log(5^A\,2^B)=C}$

$\textsf{Convert into exponential form, we get}$

$\mathsf{200^C=5^A\,2^B}$

$\mathsf{(25{\times}8)^C=5^A\,2^B}$

$\mathsf{(5^2{\times}2^3)^C=5^A\,2^B}$

$\mathsf{5^{2C}{\times}2^{3C}=5^A\,2^B}$

$\textsf{Equating powers on bothsides we get}$

$\mathsf{A=2C\;\;and\;\;B=3C}$

$\textsf{But A and B have no common factors}$

$\implies\mathsf{C=1}$

$\implies\mathsf{A=2\;\;\&\;\;B=3}$

$\mathsf{Now,}$

$\mathsf{A+B+C=2+3+1=6}$

$\implies\boxed{\mathsf{A+B+C=6}}$

$\underline{\textbf{Answer:}}$

$\mathsf{Option (2) is correct}$

Answer 2

Given  : There exist positive integers A, B and C with no common factors greater than 1, such that Alog 5 + Blog 2 = C.

 the base of the log is 200)

To Find :  The sum A+B+C

Solution:

Alog 5 + Blog 2 = C.

=> log 5^A + log 2^B = C

=> log ( 5^A * 2^B)  = C

=> 200^C  = 5^A * 2^B

=> ( 5² * 2³)^C =  5^A * 2^B

=> 5^2C * 5^3C = 5^A * 2^B

=> A = 2C

B = 3C

A , B and C are co prime

Hence  C = 1 ,  A = 2 , B = 3

A + B + C = 2 + 3 + 1  = 6

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