Question

let's see who can solve this.

Answer 1

Assume smallest number is x.

Find the least common multiple (LCM) of 468 and 520:
$520=2.2.2.5.13$ and $468=2.2.3.3.13$
$LCM=2.2.2.3.3.5.13=4680$

$x=LCM-17=4680-17=4663$

Answer 2

It's Pretty Simple, For a moment , Please understand What I am doing Carefully,

Let the least number be x,
Now, According to the question, 

x+17 is divisible by both 520 and 468,

Now, If it is divisible by 520, Then it must be a Multiple of 520, So,

Let x+17 = 520*a (Where a is any positive Integer other than 1),

If it is divisible by 468, Then it must be a multiple of 468, So,
Let x+17 = 468*b ( Where b is any positive integer other than 1),

I wrote Other than 1 Because, If a or b is 1 then it is either 468 or 520, But it can't divisible by both, So Let us find out What it is,

If a number is divisible by both x and y Then Definitely the number can also be divisible by L.C.M ( Least Common Multiple ) of x and y,

So,  x+17 is Divisible by , L.c.m of 520 and 468,

L.c.m of 520 and 468 is 4680,

So, x + 17 is Divisible by 4680, And ,
We also know that , L.c.m is least multiple that can divisible by both the numbers,

Which means, 4680 is the least number that can be divisible 520 and 468,

So therefore x+17 = 4680,
=> x = 4663,

So therefore the least number which when increased by 17 divisible by 520 and 468 is 4663,

Hope you understand, Have a great day !

Thanking you, Bunti 360 !