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.

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:

Verification / 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