When a number is divided by 512 it leaves a remainder 67. If the same number is divided by 32, then what will be the remainder?

The correct answer is Option 4: 3.

Step-by-Step Explanation:

Let the given number be $N$.

According to the division algorithm, any number $N$ can be expressed as:

$N=\text{Dividend}=(\text{Divisor}\times \text{Quotient})+\text{Remainder}$

Step 1: Express the number using the first condition

When $N$ is divided by $512$, it leaves a remainder of $67$.

Let $q$ be the quotient. Then we can write:

$N=512q+67$

Step 2: Rewrite the expression for division by 32

We need to find the remainder when $N$ is divided by $32$.

Notice that $512$ is completely divisible by $32$ ($512=32\times 16$).

Substitute $512=32\times 16$ into the expression for $N$:

$N=(32\times 16)q+67$

$N=32\left(16q\right)+67$

Now, split the remainder $67$ into a multiple of $32$ plus an extra term:

$67=(32\times 2)+3=64+3$

Substitute this back into the equation:

$N=32\left(16q\right)+32\left(2\right)+3$

$N=32(16q+2)+3$

Step 3: Determine the final remainder

The term $32(16q+2)$ is a multiple of $32$, so it is completely divisible by $32$.

Therefore, the remainder when $N$ is divided by $32$ depends only on dividing the original remainder $67$ by $32$:

$\text{Remainder}=67mod32=3$

Hence, the required remainder is 3.