To find the remainder when a number is divided by 31, we need to use the concept of modular arithmetic.

Let's say the number we are considering is 'x'. We are given that when x is divided by 775, the remainder is 188. We can represent this mathematically as:

x ≡ 188 (mod 775)

Here, '≡' represents congruence, which means that x is equivalent to 188 modulo 775.

To find the remainder when x is divided by 31, we can use the concept of modular arithmetic. We need to find a number 'r' such that:

x ≡ r (mod 31)

We can rearrange the equation x ≡ 188 (mod 775) to simplify it:

x - 188 ≡ 0 (mod 775)

Now, we can divide both sides of the equation by the highest common factor (HCF) of 31 and 775, which is 31.

(x - 188) / 31 ≡ 0 (mod 775/31)

Simplifying further:

(x - 188) / 31 ≡ 0 (mod 25)

Since 31 is relatively prime to 25, we can cancel it out:

x - 188 ≡ 0 (mod 25)

Now, to find the remainder when x is divided by 31, we need to find a number 'r' that satisfies the equation:

x - 188 ≡ r (mod 25)

To find this value, we can substitute different values for x and check which one satisfies the equation. Starting with x = 188, we have:

188 - 188 ≡ 0 (mod 25)

So, the remainder when x is divided by 31 is 0.

Therefore, the HCF of 775 and 31 is 0.