What is the value of \(x (0 \le x \le 8)\) if \((100^{97} + 100^{54} + x + 1)\) leaves a remainder 0 when divided by 9?
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.
Similar Questions
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Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
