Question:
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 is the smallest possible total number of sweets?
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 is the smallest possible total number of sweets?
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Express the total as N = 25x + 8, then find the smallest x for which N - 22 is exactly divisible by 28.
Updated On: Jul 16, 2026
- 328
- 348
- 358
- Data inadequate
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Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Write the two conditions as equations.
Let the total number of sweets be \(N\), and let each of the 25 children receive \(x\) sweets. Since 8 sweets remain after equal distribution:
\[ N = 25x + 8 \]
If instead there were 28 children, 22 sweets would remain, so \(N - 22\) must be exactly divisible by 28.
Step 2: Combine the two conditions.
Substituting, \(25x + 8 - 22 = 25x - 14\) must be divisible by 28.
Rewrite \(25x - 14 = 28x - (3x+14)\). Since \(28x\) is already divisible by 28, we need \(3x + 14\) to be divisible by 28 as well.
Testing small values of \(x\): at \(x = 14\), \(3(14) + 14 = 42 + 14 = 56 = 28 \times 2\), which works, and it's the smallest such \(x\).
Final Answer:
Total sweets \(N = 25(14) + 8 = 350 + 8 = 358\). Checking: \(358 = 28(12) + 22\), confirming the second condition too. \[ \boxed{N = 358} \]
Let the total number of sweets be \(N\), and let each of the 25 children receive \(x\) sweets. Since 8 sweets remain after equal distribution:
\[ N = 25x + 8 \]
If instead there were 28 children, 22 sweets would remain, so \(N - 22\) must be exactly divisible by 28.
Step 2: Combine the two conditions.
Substituting, \(25x + 8 - 22 = 25x - 14\) must be divisible by 28.
Rewrite \(25x - 14 = 28x - (3x+14)\). Since \(28x\) is already divisible by 28, we need \(3x + 14\) to be divisible by 28 as well.
Testing small values of \(x\): at \(x = 14\), \(3(14) + 14 = 42 + 14 = 56 = 28 \times 2\), which works, and it's the smallest such \(x\).
Final Answer:
Total sweets \(N = 25(14) + 8 = 350 + 8 = 358\). Checking: \(358 = 28(12) + 22\), confirming the second condition too. \[ \boxed{N = 358} \]
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