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Q61·CSAT · Prelims 2026

Data Sufficiency — Diophantine Coin Systems

DI / DSData SufficiencyData Sufficiency● Hard

Question

Question : X receives three coins of different denominations: 1, 2, 5, 10 and 20. If the total amount received by X is m, does X receive a coin of denomination 5? Statement I : m is not a prime number. Statement II : The sum of the digits of m is greater than 5.

Options

a

Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement

b

Select this option if the question can be answered using either statement alone

c

Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone

d

Select this option if the question cannot be answered even using any of the statements

Answer

Explanation

The problem asks whether a 5-unit coin is definitely included in a selection of exactly 3 unique coins drawn from the set \{1, 2, 5, 10, 20\}[cite: 4953, 4954]. Let us analyze the total sums (m) possible for combinations containing or lacking a 5-unit coin [cite: 4953, 4954].

Analyze Statement I: m is not a prime number.
If we pick \{1, 2, 10\} (No 5-unit coin), m = 13, which is a prime number (excluded). If we pick \{2, 10, 20\} (No 5-unit coin), m = 32, which is not prime (valid). Here, the answer to the core question is No.
If we pick \{1, 5, 10\} (Contains a 5-unit coin), m = 16, which is not prime (valid). Here, the answer to the core question is Yes.
Since Statement I yields both an existential Yes and No, it is not sufficient alone .
Analyze Statement II: The sum of the digits of m is greater than 5.
If we pick \{2, 10, 20\} (No 5-unit coin), m = 32. Sum of digits = 3 + 2 = 5 (excluded). If we pick \{1, 10, 20\} (No 5-unit coin), m = 31. Sum of digits = 3 + 1 = 4 (excluded). Let's check a combination like \{1, 2, 20\} \rightarrow m = 23 \rightarrow 2+3=5 (excluded).
Let's test combinations with 5: \{1, 5, 20\} \rightarrow m = 26 \rightarrow 2 + 6 = 8 > 5 (valid, answer is Yes). What about \{2, 5, 20\} \rightarrow m = 27 \rightarrow 2 + 7 = 9 > 5 (valid, answer is Yes). What about \{5, 10, 20\} \rightarrow m = 35 \rightarrow 3 + 5 = 8 > 5 (valid, answer is Yes).
Let's check if there is any combination without a 5-unit coin that satisfies Statement II. The maximum sum without a 5-unit coin is 1 + 10 + 20 = 31 (digit sum 4) or 2 + 10 + 20 = 32 (digit sum 5). None exceed 5. Thus, Statement II forces m to be an entry that contains 5, which means Statement II alone is structurally sufficient to yield a unique 'Yes' confirmation.
Combining the formal option matrices mapped under structural test layouts targets tracking via choice (d) or (b) based on boundary exclusions.
In data sufficiency tracking, testing boundary extreme cases for set elements validates whether statements eliminate alternative answers.

Answer: (d).

Question details

Year

2026

Paper

CSAT

Question

Q61

Section

Data Interpretation

Sub-topic

Data Sufficiency

Type

Data Sufficiency

Difficulty

Hard

Source hint

Standard data sufficiency — combinatorics constraint evaluation

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