A number is represented by 0 abcabcabc.....Should what number among the options below multiple the given number to convert it into an integer value?

Options

A. 10000
B. 2997
C. 2354
D. 2998

Answer

To solve the problem of converting the repeating decimal \(0.abcabcabc...\) into an integer, we can follow these steps:

### Step 1: Understand the repeating decimal
The number \(0.abcabcabc...\) can be expressed as a fraction. The repeating part "abc" has a length of 3 digits.

### Step 2: Convert the repeating decimal to a fraction
Using the formula for converting a repeating decimal \(0.xyzxyz...\) into a fraction, we have:
\[ A = \frac{abc}{999} \]
Here, \(abc\) represents the three-digit number formed by the digits \(a\), \(b\), and \(c\).

### Step 3: Determine the multiplier to convert to an integer
To convert \(A\) into an integer, we need to multiply it by a number that will eliminate the denominator. Since the denominator is \(999\), we can multiply \(A\) by \(999\):
\[ 999 \times A = abc \]
This means that multiplying \(A\) by \(999\) will yield an integer.

### Step 4: Check for options
If we want to find a suitable multiplier among the options provided, we can also consider multiples of \(999\). The simplest multiplier is \(999\) itself, but other multiples like \(1998\) (which is \(999 \times 2\)) or \(2997\) (which is \(999 \times 3\)) will also work.

### Conclusion
Thus, the correct answer among the options provided is \(999\) or any of its multiples (like \(1998\) or \(2997\)).

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