Least number with uniform remainder: Find the least positive number which, when divided by 15, 27, 35, and 42, leaves a remainder of 7 in each case.
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A1883
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B1897
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C1987
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D2007
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E1890
Answer
Correct Answer: 1897
Explanation
Introduction / Context:When a number leaves the same remainder r upon division by several moduli, then that number minus r is a common multiple of the moduli. This reduces directly to computing an LCM and then adding the remainder.
Given Data / Assumptions:
- N ≡ 7 (mod 15)
- N ≡ 7 (mod 27)
- N ≡ 7 (mod 35)
- N ≡ 7 (mod 42)
Concept / Approach:Let M be LCM(15, 27, 35, 42). Then the least positive N is M + 7. Compute M using prime powers: take the highest power of each prime across the factorizations.
Step-by-Step Solution:
Prime factors: 15 = 3 * 5; 27 = 3^3; 35 = 5 * 7; 42 = 2 * 3 * 7LCM = 2^1 * 3^3 * 5^1 * 7^1 = 2 * 27 * 5 * 7 = 1890Therefore, least N = 1890 + 7 = 1897Verification / Alternative check:Check: 1897 mod 15 = 7; mod 27 = 7; mod 35 = 7; mod 42 = 7. All satisfied.
Why Other Options Are Wrong:1883, 1987, and 2007 do not satisfy all four congruences. 1890 is LCM itself and would leave remainder 0, not 7.
Common Pitfalls:Leaving out the factor 2 from 42 when computing LCM, or forgetting to add the remainder after finding the LCM.
Final Answer:1897