After distributing the sweets equally among 25 children, 8 sweets remain. Had the number of children been 28, 22 sweets would have been left after equally distributing. What was the total number of sweets?

## Concept & Logic This problem tests the application of the Remainder Theorem and modular arithmetic. We are looking for a number $N$ that leaves specific remainders when divided by two different divisors. $$ N = D_1 \times Q_1 + R_1 $$ $$ N = D_2 \times Q_2 + R_2 $$ ## Step-by-Step Solution * **Given:** Let the total number of sweets be $N$. Condition 1: $N \div 25$ leaves a remainder of 8. So, $N = 25a + 8$. Condition 2: $N \div 28$ leaves a remainder of 22. So, $N = 28b + 22$. * **Deduction:** We can test the given options to see if they satisfy both conditions. * **Calculation:** Let's test Option (c) 358: $358 \div 25 = 14$ with a remainder of 8. (Matches Condition 1) $358 \div 28 = 12$ with a remainder of $358 - (28 \times 12) = 358 - 336 = 22$. (Matches Condition 2) * **Deduction:** 358 works perfectly. However, is it the *only* answer? According to the Chinese Remainder Theorem, the general solution for such a system is $N = N_0 + k \times LCM(D_1, D_2)$, where $N_0$ is the smallest positive solution. $LCM(25, 28) = 700$. So, any number in the form $N = 358 + 700k$ (where $k$ is an integer $\ge 0$) will satisfy both conditions. Valid numbers are 358, 1058, 1758, 2458, etc. * **Conclusion:** Because no upper limit or exact constraint is given for the total number of sweets, there are infinitely many valid answers. We cannot determine the exact total. ## Exam Strategy & Shortcut When a question asks "What was the *total* number..." (implying a single unique absolute value) but only provides remainder conditions without an upper bound or a quotient relationship, immediately suspect "Data inadequate". Check the options. While 358 works, the structural lack of constraints means the answer cannot be uniquely determined. ## Common Pitfall The most common trap is testing the options, finding that 358 works perfectly for both conditions, and immediately marking (c) as the answer. Examiners include 358 specifically to bait students who don't realize that multiple solutions exist (like 1058) and thus the data is technically inadequate to find *the* unique total. ## Final Answer Therefore, the correct answer is **Data inadequate**.