中国剩余定理1（Chinese remainder theorem）

If we add three more soldiers, it will become: "there's nothing left in a group of 4,5,7"

So, $4\times5\times7=140$ satisfies the above. Subtract by 3 and get the answer=137

Relevant wiki: Chinese Remainder Theorem

Let the least number of soldiers be $n$ . Then

$\begin{aligned} n & \equiv 1 \text{ (mod 4)} \\ \implies n & \equiv 4a + 1 & \small \color{#3D99F6} \text{where }a \text{ is an integer.} \\ 4a + 1 & \equiv 2 \text{ (mod 5)} \\ 4a & \equiv 1 \text{ (mod 5)} \\ a & \equiv 4 \\ \implies n & \equiv 4\times 5 b + 4a+1 = 20b+17 & \small \color{#3D99F6} \text{where }b \text{ is an integer.} \\ 20b + 17 & \equiv 4 \text{ (mod 7)} \\ 6b + 3 & \equiv 4 \text{ (mod 7)} \\ 6b & \equiv 1 \text{ (mod 7)} \\ b & \equiv 6 \text{ (mod 7)} \\ \implies n & \equiv 20(6)+17 = \boxed{137} \end{aligned}$