Question

Find two natural numbers whose sum is 85 and the least common multiple is 102.

Answer 1

let first number is= x
other wll be =85-x
x*(85-x)=102
solve it further

Answer 2

it dont think it gives integers

Answer 3

http://www.wolframalpha.com/input/?i=x*%2885-x%29%3D102

Answer 4

factorise 102

Answer 5

that will be 2,3,17

Answer 6

102 = 2*3*17

17X + 6Y = 85 ( no solution coz 17 is a factor of 85 and 6 is not a factor 17)
34X + 3Y = 85 (no solution)
51X + 2Y = 85 (X = 1 and Y = 17)

so , numbers are 51 and 34

Answer 7

For a general method without bruteforce, you may try something like below :

Say the required two numbers are \(A\) and \(B\)

We're given :
\[\large A+B = 85 = 5\times 17\tag{1}\]
\[LCM(A,B) = 102 = 2\times 3\times 17 \]


\(\gcd(A,B) \) divides both \(A\) and \(B\), it has to divide \(\large A+B\) also; that makes \(\gcd(A,B) = 17\)


using identity : \(A*B = \gcd(A,B) \times LCM(A,B) \), we have :
\(\large AB =17\times 102 \tag{2}\)

you can solve \((1), (2)\)