Understanding the Remainder Problem

We are given a number that leaves a specific remainder when divided by a larger number. We need to find the remainder when this same number is divided by a smaller number.

Mathematical Representation

Let the number be denoted by $N$. According to the problem statement:

• When $N$ is divided by 8379, the remainder is 87.

This can be written using the division algorithm as:

$N = 8379 \times q + 87$, where $q$ is the quotient.

Finding the Remainder with 63

We need to find the remainder when $N$ is divided by 63. This means we need to calculate $N \pmod{63}$.

Substitute the expression for $N$:

$N \pmod{63} = (8379 \times q + 87) \pmod{63}$

Using the properties of modular arithmetic, we can write:

$N \pmod{63} = \left( (8379 \pmod{63}) \times (q \pmod{63}) + (87 \pmod{63}) \right) \pmod{63}$

Calculating Intermediate Remainders

First, find the remainder when 8379 is divided by 63:

$ 8379 \div 63 = 133 \text{ with a remainder of } 0 $

So, $8379 \pmod{63} = 0$.

Next, find the remainder when 87 is divided by 63:

$ 87 \div 63 = 1 \text{ with a remainder of } 24 $

So, $87 \pmod{63} = 24$.

Determining the Final Remainder

Now substitute these calculated remainders back into the expression for $N \pmod{63}$:

$N \pmod{63} = \left( (0) \times (q \pmod{63}) + 24 \right) \pmod{63}$

$N \pmod{63} = (0 + 24) \pmod{63}$

$N \pmod{63} = 24 \pmod{63}$

$N \pmod{63} = 24$

The remainder when the same number is divided by 63 is 24.