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.
- (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement ✓ UPSC's answer
- (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
Why the answer is (a)
• X receives three coins from the set {1, 2, 5, 10, 20}, so the total amount m is the sum of three distinct denominations.
• Statement I states m is not a prime number. The possible sums are: 1+2+5=8, 1+2+10=13, 1+2+20=23, 1+5+10=16, 1+5+20=26, 1+10+20=31, 2+5+10=17, 2+5+20=27, 2+10+20=32, 5+10+20=35.
• The prime sums are 13, 23, 31, 17. The non-prime sums are 8, 16, 26, 27, 32, 35.
• In all non-prime cases (8, 16, 26, 27, 32, 35), the coin of denomination 5 is present (8=1+2+5, 16=1+5+10, 26=1+5+20, 27=2+5+20, 32=2+10+20? No, 32=2+10+20 does not have 5. Wait. Let's re-evaluate 32. 2+10+20=32. This sum does NOT contain 5. So Statement I alone is insufficient? Let's re-read the options and key. Key is (a). This implies one statement is sufficient. Let's re-check Statement II.
• Statement II states the sum of digits of m is greater than 5. Let's check the digit sums for all possible m values: 8 (sum 8>5, has 5), 13 (sum 4, no), 23 (sum 5, no), 16 (sum 7>5, has 5), 26 (sum 8>5, has 5), 31 (sum 4, no), 17 (sum 8>5, has 5? 2+5+10=17. Yes, has 5), 27 (sum 9>5, has 5), 32 (sum 5, no), 35 (sum 8>5, has 5).
• Let's list m and 'has 5?':
8 (1,2,5): Yes. Digit sum 8 > 5.
13 (1,2,10): No. Digit sum 4 <= 5.
23 (1,2,20): No. Digit sum 5 <= 5.
16 (1,5,10): Yes. Digit sum 7 > 5.
26 (1,5,20): Yes. Digit sum 8 > 5.
31 (1,10,20): No. Digit sum 4 <= 5.
17 (2,5,10): Yes. Digit sum 8 > 5.
27 (2,5,20): Yes. Digit sum 9 > 5.
32 (2,10,20): No. Digit sum 5 <= 5.
35 (5,10,20): Yes. Digit sum 8 > 5.
• Statement II (digit sum > 5) corresponds to m in {8, 16, 26, 17, 27, 35}. In ALL these cases, the coin 5 is present. Thus, Statement II alone is sufficient to answer 'Yes'.
• Statement I (m is not prime) corresponds to m in {8, 16, 26, 27, 32, 35}. Here, m=32 (2+10+20) does not have coin 5, while others do. So Statement I is insufficient.
• Therefore, the question can be answered using Statement II alone, but not Statement I alone. This matches option (a).
Why the other options are wrong
- (b) Select this option if the question can be answered using either statement alone
- Statement I is insufficient because m=32 (sum of 2, 10, 20) is not prime but does not include the coin of denomination 5, whereas other non-prime sums do.
- (c) Select this option if the question can be answered using both the statements together…
- The question can be answered using Statement II alone, so it is not necessary to use both statements together.
- (d) Select this option if the question cannot be answered even using any of the statements
- The question can be definitively answered using Statement II alone, as all sums with a digit sum greater than 5 include the coin of denomination 5.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.