Sort Of Divisible by All Four Numbers

Let $x$ be a positive integer that leaves a remainder of 4, when divided by 6, 9, 15 and 18.

Then $x-4$ must be divisible by 6, 9, 15 and 18.

By the definition of lowest common multiple , we have $x-4$ is a multiple of $\text{lcm}(6,9,15,18)$ .

Since $6 = 2\times3$ , $9 =3^2$ , $15 = 3\times5$ , $18 = 2\times3^2$ , then $\text{lcm}(6,9,15,18) = 2\times3^2 \times5 = 90$ .

In other words, the possible values of $x-4$ are $90, 180, 270, 360 , 450, \ldots$ .
Or equivalently, the possible values of $x$ are $94, 184,274,364,454,\ldots$ .

We want to find the minimum value of $x$ that is also a multiple of 7. Looking through the list of numbers above shows that $\boxed{364}$ is the number we are looking for.