Which is the smallest multiple of 7, which leaves 5 as the remainder in each case. When divided by 8, 9, 12 and 15?

The correct option is 1,085.

Let us find the solution step-by-step.

Step 1: Understand the requirements of the problem.
We need to find the smallest number, say $N$, such that:
1. $N$ is a multiple of 7 (i.e., $N$ is completely divisible by 7).
2. When $N$ is divided by 8, 9, 12, and 15, it leaves a remainder of 5 in each case.

Step 2: Find the Least Common Multiple (LCM) of the divisors.
Any number that leaves a remainder of 5 when divided by 8, 9, 12, and 15 must be of the form:
$N=\left(\text{LCM of 8, 9, 12, 15}\right)\times k+5$
where $k$ is a positive integer.

Let us find the LCM of 8, 9, 12, and 15 using prime factorization:
- Prime factorization of 8: $2^3$
- Prime factorization of 9: $3^2$
- Prime factorization of 12: $2^2\times 3$
- Prime factorization of 15: $3\times 5$

The LCM is the product of the highest power of each prime factor present in these numbers:
$\text{LCM}=2^3\times 3^2\times 5$
$\text{LCM}=8\times 9\times 5=360$

So, the required number must be of the form:
$N=360k+5$

Step 3: Find the value of $k$ such that $N$ is divisible by 7.
We need to find the smallest integer value of $k$ for which $360k+5$ is divisible by 7.
Let us express 360 in terms of a multiple of 7:
$360=7\times 51+3$
Therefore, we can rewrite the expression as:
$360k+5=(7\times 51+3)k+5$
$360k+5=7\times 51k+(3k+5)$

For the number to be divisible by 7, the term $3k+5$ must be divisible by 7.
Let us test positive integer values of $k$ starting from 1:
- If $k=1$: $3\left(1\right)+5=8$ (not divisible by 7)
- If $k=2$: $3\left(2\right)+5=11$ (not divisible by 7)
- If $k=3$: $3\left(3\right)+5=14$ (which is divisible by 7 since $14/7=2$)

Thus, the smallest integer value is $k=3$.

Step 4: Calculate the final number.
Substitute $k=3$ back into our expression for $N$:
$N=360\times 3+5$
$N=1080+5=1085$

Thus, the smallest multiple of 7 which leaves a remainder of 5 in each case when divided by 8, 9, 12, and 15 is 1,085.