Finding the Greatest Divisor with Remainders

The question asks for the largest number that divides $135$ leaving a remainder of $5$, and divides $281$ leaving a remainder of $1$. This means the number we are looking for is the Greatest Common Divisor (GCD) of the numbers obtained after subtracting the remainders from the original numbers.

Adjusting the Numbers

First, adjust the numbers by subtracting their respective remainders:

• For $135$ with remainder $5$: $135 - 5 = 130$.
• For $281$ with remainder $1$: $281 - 1 = 280$.

The greatest number we need is the GCD of $130$ and $280$.

Calculating the GCD of 130 and 280

We find the prime factorization of each adjusted number:

• Prime factorization of $130$: $130 = 10 \times 13 = (2 \times 5) \times 13$
• Prime factorization of $280$: $280 = 10 \times 28 = (2 \times 5) \times (4 \times 7) = (2 \times 5) \times (2^2 \times 7) = 2^3 \times 5 \times 7$

To find the GCD, we take the lowest power of each common prime factor:

• Common prime factors are $2$ and $5$.
• The lowest power of $2$ is $2^1$.
• The lowest power of $5$ is $5^1$.

Therefore, the GCD is:

$ \text{GCD}(130, 280) = 2^1 \times 5^1 = 10 $

Result

The greatest number that satisfies the conditions is $10$.