Question

Question #2f846

Answer 1

`color(green)("Totally Revised method")`

`color(blue)(x= 535)`

Explanation:

Let the divisor be `x`
Let the first number be `f`
Let the second number be `s`
Let a, b & c be integer values

`color(blue)("Condition 1 with remainder 473")`

`f/x = a+ 473/x` ..............................(1)

`color(blue)("Condition 2 with remainder 298")`

`s/x = b+298/x`...............................(2)

`color(blue)("Condition 3 with remainder 236")`

`(f+s)/x=c+236/x`................................(3) `color(red)("corrected")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I think I was initially using a very wrong view point to solve this:

`color(blue)("New method")`

From (1) we know that `x>473`

If we add the remainders from (1) and (2) then we have 771 but the division by `x` reduces this to 236. As in (3)#

From this we can deduce that `473 < x < 771`

As the remainder in (3) is 236 then it has to mean that:

`color(white)(xxxxxxxx)x=771-236 = 535`

Answer 2

`535`

Explanation:

When first number is divided by a divisor `x` and remainder is `=473`
`=>` divisor `x>473` ...(1)
Similarly for the second number we have `x>298` ......(2)
We observe if (1) is satisfied then (2) gets satisfied automatically as `473>298`

When sum of two numbers is divided by same divisor `x` and we have remainder `236`
it is equivalent to division of sum of two remainders by same divisor `x`
Hence we have the connecting expression
`(473+298)=nx+236` .....(3)
where `n` is an integer and `x` must satisfy equations (1).

For `n=1` we have
`x=(771-236)=535`

For `n=2`
`x=535/2=267.5`
for `n=2,3....`
the obtained values of `x` do not satisfy equation (1)
Hence only valid value is `=535`