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11 Aug 2025, 09:16
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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Let m be the 1,003rd digit to the right of the decimal point in the repeated decimal expansion 0.13579, and let n equal the 78th digit to the right of the decimal point in the repeated decimal expansion 0.2468. What is the value of \(m^n\)?
24 Aug 2025, 06:04
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Official Explanation
The bar above the digits to the right of the decimal point means that those digits repeat forever. Begin to write the first sequence and look for a pattern: .135791357913579... . Notice that the 5th, 10th, and 15th digits are all 9, suggesting that any digit that is a multiple of 5 to the right of the decimal will equal 9. Then the 1,000th digit will be a 9, and the 1,003rd digit will be three digits after this, a 5. In the other number, 0.246824682468 ..., the pattern contains four digits before repeating. Notice that the 4th, 8th, and 12th digits are all 8. We again see that any multiple of the pattern length will be the last digit of the pattern—that is, any digit a multiple of 4 to the right of the decimal point will be an 8. This means that the 76th digit is an 8, because 76 is a multiple of 4, since 76 = 19 × 4. Then the 78th digit will be two digits after this, a 4. Then mn equals 54 = 625.
Answer: 625
The bar above the digits to the right of the decimal point means that those digits repeat forever. Begin to write the first sequence and look for a pattern: .135791357913579... . Notice that the 5th, 10th, and 15th digits are all 9, suggesting that any digit that is a multiple of 5 to the right of the decimal will equal 9. Then the 1,000th digit will be a 9, and the 1,003rd digit will be three digits after this, a 5. In the other number, 0.246824682468 ..., the pattern contains four digits before repeating. Notice that the 4th, 8th, and 12th digits are all 8. We again see that any multiple of the pattern length will be the last digit of the pattern—that is, any digit a multiple of 4 to the right of the decimal point will be an 8. This means that the 76th digit is an 8, because 76 is a multiple of 4, since 76 = 19 × 4. Then the 78th digit will be two digits after this, a 4. Then mn equals 54 = 625.
Answer: 625
