Arithmetic Word Problem — Remainders and the Chinese Remainder Idea After distributing sweets equally among 25 children, 8 sweets remain. If the number of children had been 28, then 22 sweets would have remained after equal distribution. Find the total number of sweets.

Introduction / Context:
This is a classic modular arithmetic problem involving remainders. We are given two different group sizes (25 and 28) and the remainders of sweets left over in each case. The task is to determine the total number of sweets consistent with both conditions.

Given Data / Assumptions:

• Total sweets = S (a whole number).
• When divided by 25, remainder is 8 → S leaves 8 after equal sharing among 25 children.
• When divided by 28, remainder is 22 → S leaves 22 after equal sharing among 28 children.

Concept / Approach:
Use modular arithmetic. The conditions translate to two congruences:
S mod 25 = 8S mod 28 = 22We can solve by inspection, systematic search, or using the Chinese Remainder approach because 25 and 28 are coprime (gcd = 1).

Step-by-Step Solution:
1) From S mod 25 = 8 ⇒ S = 25k + 8 for some integer k.2) Substitute into the second condition: 25k + 8 ≡ 22 (mod 28) ⇒ 25k ≡ 14 (mod 28).3) Reduce 25k mod 28: since 25 ≡ -3 (mod 28), we have -3k ≡ 14 (mod 28) ⇒ 3k ≡ -14 ≡ 14 (mod 28).4) Solve 3k ≡ 14 (mod 28). The inverse of 3 mod 28 is 19 (since 319 = 57 ≡ 1 mod 28). Thus k ≡ 1419 = 266 ≡ 266 - 928 = 266 - 252 = 14 (mod 28).5) Take k = 14 (smallest positive). Then S = 2514 + 8 = 350 + 8 = 358.

Verification / Alternative check:
Check S mod 25: 358 = 2514 + 8 ⇒ remainder 8 ✔Check S mod 28: 358 = 2812 + 22 ⇒ remainder 22 ✔Thus S = 358 fits both conditions.

Why Other Options Are Wrong:

• 328: 328 mod 25 = 3, not 8; fails first condition.
• 348: 348 mod 28 = 12, not 22; fails second condition.
• Data inadequate: The two coprime moduli give a unique solution modulo 25*28 = 700; data are sufficient.

Common Pitfalls:
Subtracting the remainders directly or guessing. Always translate to precise congruences and solve systematically to avoid errors.

Final Answer:
358