Understanding the Remainder Problem

We need to determine the value of \(x\), given that \(0 \le x \le 8\). The condition is that the expression \((100^{97} + 100^{54} + x + 1)\) leaves a remainder of 0 when divided by 9. This means the expression must be exactly divisible by 9.

Applying Modular Arithmetic

To find the unknown value \(x\), we can analyze the remainders of the terms in the expression when divided by 9. This technique is known as modular arithmetic.

Step 1: Find the remainder of 100 when divided by 9.

When we divide 100 by 9, we get:

\(100 = 11 \times 9 + 1\)

So, the remainder is 1. In modular arithmetic, we express this as:

\(100 \equiv 1 \pmod{9}\)

Step 2: Calculate the remainders for the powers of 100.

We can use the property that if \(a \equiv b \pmod{m}\), then \(a^n \equiv b^n \pmod{m}\).

For the term \(100^{97}\):

\(100^{97} \pmod{9} \equiv (100 \pmod{9})^{97} \pmod{9}\)

Since \(100 \equiv 1 \pmod{9}\), we have:

\(100^{97} \equiv 1^{97} \pmod{9}\)

\(100^{97} \equiv 1 \pmod{9}\)

Similarly, for the term \(100^{54}\):

\(100^{54} \pmod{9} \equiv (100 \pmod{9})^{54} \pmod{9}\)

\(100^{54} \equiv 1^{54} \pmod{9}\)

\(100^{54} \equiv 1 \pmod{9}\)

Step 3: Determine the remainder of the full expression.

Now, we find the remainder of the entire expression \((100^{97} + 100^{54} + x + 1)\) when divided by 9. We use the property that the remainder of a sum is the sum of the individual remainders (modulo 9).

\((100^{97} + 100^{54} + x + 1) \pmod{9}\)

Substitute the remainders we found:

\( \equiv (1 + 1 + x + 1) \pmod{9} \)

Simplify the expression:

\( \equiv (3 + x) \pmod{9} \)

Solving for x

The problem states that the expression has a remainder of 0 when divided by 9. Therefore:

\( (3 + x) \equiv 0 \pmod{9} \)

This equation implies that the sum \((3 + x)\) must be a multiple of 9. The multiples of 9 are \(..., -9, 0, 9, 18, 27, ...\).

We are given the constraint that \(x\) must be between 0 and 8, inclusive (\(0 \le x \le 8\)). Let's examine the possible values of \((3 + x)\) within this range:

• If \(x=0\), \(3+x = 3\).
• If \(x=1\), \(3+x = 4\).
• If \(x=2\), \(3+x = 5\).
• If \(x=3\), \(3+x = 6\).
• If \(x=4\), \(3+x = 7\).
• If \(x=5\), \(3+x = 8\).
• If \(x=6\), \(3+x = 9\).
• If \(x=7\), \(3+x = 10\).
• If \(x=8\), \(3+x = 11\).

Among these possible sums, only 9 is a multiple of 9.

Therefore, we must have:

\( 3 + x = 9 \)

To find \(x\), we subtract 3 from both sides:

\( x = 9 - 3 \)

\( x = 6 \)

This value of \(x=6\) satisfies the given condition \(0 \le x \le 8\). Thus, the required value of \(x\) is 6.