The least number which when divided by $15,27,35$ and $42$ leaves a remainder of $7$ in each division is

The general structural form for such an integer is:
$$\text{Required Number}=\text{LCM}(15,27,35,42)+\text{Remainder}$$

Let's find the Least Common Multiple ($\text{LCM}$) by prime factorization:

* $15=3\times 5$
* $27=3^3$
* $35=5\times 7$
* $42=2\times 3\times 7$

Taking the maximum power of each prime factor involved:

$$\text{LCM}=2^1\times 3^3\times 5^1\times 7^1=2\times 27\times 5\times 7=1890$$

Now, adding the common remainder constant ($7$):

$$\text{Required Number}=1890+7=1897$$