Question:medium
A number \( x = 3.1212121212\ldots \). By what least number should we multiply \( x \), so that we obtain an integer?
A number \( x = 3.1212121212\ldots \). By what least number should we multiply \( x \), so that we obtain an integer?
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For repeating decimals, express the decimal part as a fraction to simplify the problem.
Updated On: Mar 20, 2026
- 3
- 9
- 30
- 33
- 99
Show Solution
The Correct Option is D
Solution and Explanation
Step 1: Express the number as a fraction.
Define \( x = 3.1212121212\ldots \), a repeating decimal. Express this as: \[x = 3 + 0.1212121212\ldots\] Let \( y = 0.1212121212\ldots \). Then: \[y = 0.\overline{12}.\] Step 2: Convert the repeating decimal to a fraction.
To convert \( y = 0.\overline{12} \) to a fraction, multiply by 100: \[100y = 12.\overline{12}.\] Subtract \( y = 0.\overline{12} \) from \( 100y = 12.\overline{12} \): \[100y - y = 12.\overline{12} - 0.\overline{12},\] \[99y = 12,\] \[y = \frac{12}{99} = \frac{4}{33}.\] Step 3: Combine the integer and the fraction.
Therefore: \[x = 3 + \frac{4}{33} = \frac{99}{33} + \frac{4}{33} = \frac{103}{33}.\] Step 4: Find the least multiplier to make \( x \) an integer.
To make \( x = \frac{103}{33} \) an integer, multiply the numerator and denominator by 33: \[x = \frac{103 \times 33}{33 \times 33} = \frac{3399}{1089}.\] Thus, multiplying by 33 results in an integer.
Define \( x = 3.1212121212\ldots \), a repeating decimal. Express this as: \[x = 3 + 0.1212121212\ldots\] Let \( y = 0.1212121212\ldots \). Then: \[y = 0.\overline{12}.\] Step 2: Convert the repeating decimal to a fraction.
To convert \( y = 0.\overline{12} \) to a fraction, multiply by 100: \[100y = 12.\overline{12}.\] Subtract \( y = 0.\overline{12} \) from \( 100y = 12.\overline{12} \): \[100y - y = 12.\overline{12} - 0.\overline{12},\] \[99y = 12,\] \[y = \frac{12}{99} = \frac{4}{33}.\] Step 3: Combine the integer and the fraction.
Therefore: \[x = 3 + \frac{4}{33} = \frac{99}{33} + \frac{4}{33} = \frac{103}{33}.\] Step 4: Find the least multiplier to make \( x \) an integer.
To make \( x = \frac{103}{33} \) an integer, multiply the numerator and denominator by 33: \[x = \frac{103 \times 33}{33 \times 33} = \frac{3399}{1089}.\] Thus, multiplying by 33 results in an integer.
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