Finding the Remainder of Number N

The question asks for the remainder when a number N is divided by 98, given that when N is increased by 17, the result is divisible by 392.

Expressing the Problem Mathematically

We are given that N + 17 is divisible by 392. This means that N + 17 can be written as a multiple of 392. Let's use 'k' to represent any integer. We can write the relationship as:

\( N + 17 = 392 \times k \)

From this equation, we can express N in terms of k:

\( N = 392 \times k - 17 \)

Relating the Divisors: 392 and 98

We need to find the remainder when N is divided by 98. This is equivalent to finding \( N \pmod{98} \). Let's look at the relationship between 392 and 98.

We can divide 392 by 98:

\( 392 \div 98 \)

Let's perform the division:

\( 98 \times 1 = 98 \)

\( 98 \times 2 = 196 \)

\( 98 \times 3 = 294 \)

\( 98 \times 4 = 392 \)

So, 392 is exactly 4 times 98. This means 392 is a multiple of 98.

\( 392 = 4 \times 98 \)

Calculating the Remainder of N by 98

We want to find \( N \pmod{98} \). We know \( N = 392 \times k - 17 \). Let's substitute the expression for N:

\( N \pmod{98} = (392 \times k - 17) \pmod{98} \)

Since 392 is a multiple of 98 (\(392 = 4 \times 98\)), any multiple of 392 is also a multiple of 98. Therefore, \( 392 \times k \) is a multiple of 98 for any integer k.

In modular arithmetic, this means:

\( (392 \times k) \pmod{98} = 0 \)

Now, substitute this back into the expression for \( N \pmod{98} \):

\( N \pmod{98} = (0 - 17) \pmod{98} \)

\( N \pmod{98} = -17 \pmod{98} \)

To find the remainder of a negative number, we add the modulus (98) until we get a non-negative result:

\( -17 + 98 = 81 \)

So, the remainder when N is divided by 98 is 81.

Verification with an Example

Let's take a simple case where \(k=1\).

\( N + 17 = 392 \times 1 = 392 \)

\( N = 392 - 17 = 375 \)

Now, let's divide 375 by 98:

\( 375 = 98 \times q + r \)

\( 375 \div 98 \)

\( 375 = 3 \times 98 + 81 \)

Because \( 3 \times 98 = 294 \) and \( 375 - 294 = 81 \). The remainder is indeed 81.

This confirms our calculation using modular arithmetic.

The remainder when N is divided by 98 is 81.