Find the least subtraction for a uniform remainder condition What least number must be subtracted from 1294 so that when the result is divided by 9, 11, and 13 it leaves remainder 6 in each case?

Introduction / Context:
We need the smallest nonnegative integer x such that N' = 1294 - x leaves the same remainder 6 upon division by 9, 11, and 13. This is a simultaneous congruence question that reduces to using the least common multiple (LCM) of the moduli.

Given Data / Assumptions:

• N' mod 9 = 6, N' mod 11 = 6, N' mod 13 = 6.
• All moduli 9, 11, 13 are pairwise coprime.

Concept / Approach:
If N' leaves remainder 6 with each of 9, 11, 13, then N' - 6 is divisible by all three. Hence N' - 6 is a multiple of LCM(9, 11, 13). Because the moduli are pairwise coprime, the LCM is their product.

Step-by-Step Solution:
LCM(9, 11, 13) = 9 * 11 * 13 = 1287.So N' ≡ 6 (mod 1287). The closest number to 1294 that is congruent to 6 mod 1287 is 1287 + 6 = 1293.Thus 1294 - x = 1293 ⇒ x = 1.

Verification / Alternative check:
1293 ÷ 9 leaves 6; 1293 ÷ 11 leaves 6; 1293 ÷ 13 leaves 6. All checks pass.

Why Other Options Are Wrong:
0 gives 1294, which does not leave remainder 6 for all three. 2 or 3 land at 1292 or 1291, which fail at least one modulus. 7 similarly fails.

Common Pitfalls:
Forgetting to convert “same remainder 6” into the condition N' - 6 is divisible by each modulus, or using gcd instead of LCM.

Final Answer:
1