The Puzzle That Looked Like It Had One Answer
You are teaching reasoning to Class VIII in a KVS school. The period is 30 minutes, and there are 45 students. You give a data-sufficiency item: “How many students are in a class? Statement I: The teacher can divide the class into groups of 6 with none left over. Statement II: The teacher can divide the class into groups of 8 with none left over.” The options are: (A) I alone is sufficient, (B) II alone is sufficient, (C) both together are sufficient, (D) both together are not sufficient. After 6 minutes, 31 students choose (C), 9 choose (D), and 5 choose (A). One student explains, “6 and 8 have LCM 24, so the answer is 24; that is why both are needed.” Another says, “But the class could be 48.” The bell is in 10 minutes. You have a board and no projector. You want students to understand the meaning of “sufficient,” not just memorize the LCM.
1. What is the correct option, and why is the student’s LCM reasoning incomplete?
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The correct option is (D) both together are not sufficient. The student has calculated the smallest possible class size correctly: any number divisible by both 6 and 8 must be divisible by 24. But 24 is not the only possibility. The class could be 24, 48, 72, 96, and so on. “Sufficient” in data-sufficiency questions means the statements determine one unique value. Here, together they determine a set of possible values, not a single number. Statement I alone gives multiples of 6; Statement II alone gives multiples of 8; together they give multiples of 24. Because more than one class size satisfies all the information, the answer is (D), not (C). The student’s error is treating “the LCM is 24” as “the class must be 24.” The LCM gives the smallest common multiple, not a unique solution.
2. Give a five-minute board explanation that uses the students’ own numbers to correct the misconception.
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Draw two sets on the board. Under Statement I, list multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48… Under Statement II, list multiples of 8: 8, 16, 24, 32, 40, 48… Ask students to circle the numbers that appear in both lists: 24, 48, 72… Then ask: “If I tell you both statements are true, can you tell me the exact class size?” The class should say no, because 24 and 48 both work. Write the definition of sufficient in large letters: one definite answer. Then show a contrast: if the question added “the class has fewer than 30 students,” then 24 would be unique, and (C) would be correct. This contrast is the key; it separates “both statements are consistent with a value” from “both statements determine the value.”
3. What exit task would test transfer without repeating the same 6-and-8 example?
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Give a parallel item with different numbers: “How many books are in a box? Statement I: The books can be packed in packets of 4 with none left over. Statement II: The books can be packed in packets of 6 with none left over. Statement III: The box contains fewer than 30 books.” Ask students to decide whether I and II alone are sufficient, and whether adding III makes them sufficient. The intended reasoning: I and II together allow multiples of 12 - 12, 24, 36… - so they are not sufficient alone. Adding III narrows the possibilities to 12 and 24, so even with III there are two values; the answer is still not unique. This tests the definition of sufficiency rather than the LCM 24. A second, shorter item can change III to “fewer than 20 books” so that 12 becomes unique; comparing the two shows why the extra condition matters.