Question:
N = abc is a three-digit number, where a, b and c are its digits. N is a perfect square of an even number. To find N, which of the following information(s) is/are necessary and sufficient?
A. a, b, c are three consecutive digits but not in order.
B. N is divisible by 18.
C. The units digit of \(N^2\) is c.
N = abc is a three-digit number, where a, b and c are its digits. N is a perfect square of an even number. To find N, which of the following information(s) is/are necessary and sufficient?
A. a, b, c are three consecutive digits but not in order.
B. N is divisible by 18.
C. The units digit of \(N^2\) is c.
A. a, b, c are three consecutive digits but not in order.
B. N is divisible by 18.
C. The units digit of \(N^2\) is c.
Show Hint
N must be a three digit square of an even number. List all such squares, then test which single clue or pair of clues leaves only one candidate.
Updated On: Jul 13, 2026
- Only A and B together are sufficient
- A and C together are sufficient
- Only B and C together are sufficient
- Either A and C together or B and C together are sufficient
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: List the possible values of N.
N is a three digit perfect square of an even number. Even numbers from 10 to 30 give three digit squares: \(10^2=100, 12^2=144, 14^2=196, 16^2=256, 18^2=324, 20^2=400, 22^2=484, 24^2=576, 26^2=676, 28^2=784, 30^2=900\).
So N could be any of these 11 numbers unless we get more clues.
Step 2: Check statement A alone.
Statement A says a, b and c are three consecutive digits, written in some order that is not sorted. Going through the list, only 324 (digits 3, 2, 4, the set of digits 2, 3, 4) and 576 (digits 5, 7, 6, the set of digits 5, 6, 7) have three consecutive digits. So A alone leaves two candidates, 324 and 576, and cannot fix N by itself.
Step 3: Check statement B alone.
Statement B says N is divisible by 18. From the list, 144, 324, 576 and 900 are all divisible by 18. That is four candidates, so B alone is not enough either.
Step 4: Check statement C alone.
Statement C says the units digit of \(N^2\) equals c, the units digit of N. This is true for several of the 11 numbers, so C alone also fails to pin down one value.
Step 5: Combine A and C.
A narrows N to 324 or 576. Check clue C on these two: \(324^2 = 104976\), units digit 6, but c = 4 for 324, so 324 fails. \(576^2 = 331776\), units digit 6, and c = 6 for 576, so 576 passes. A and C together fix N = 576, so this pair is sufficient.
Step 6: Combine B and C.
B narrows N to 144, 324, 576 or 900. Checking clue C the same way, N = 576 satisfies both the divisibility by 18 and the units digit condition together, so B and C also settle on N = 576. This pair is sufficient too.
Step 7: Rule out the other options.
A and B together only narrow N to 324 or 576, so A and B alone do not fix N, which rules out choice (1). Since B and C alone are also sufficient, choice (2), which claims only A and C work, is too narrow. Because BOTH the pair A, C and the pair B, C independently pin down N = 576, the answer must allow either pairing.
Final Answer:
Either A and C together, or B and C together, are each enough to fix N = 576. \[ \boxed{\text{Either A and C together, or B and C together, are sufficient}} \]
N is a three digit perfect square of an even number. Even numbers from 10 to 30 give three digit squares: \(10^2=100, 12^2=144, 14^2=196, 16^2=256, 18^2=324, 20^2=400, 22^2=484, 24^2=576, 26^2=676, 28^2=784, 30^2=900\).
So N could be any of these 11 numbers unless we get more clues.
Step 2: Check statement A alone.
Statement A says a, b and c are three consecutive digits, written in some order that is not sorted. Going through the list, only 324 (digits 3, 2, 4, the set of digits 2, 3, 4) and 576 (digits 5, 7, 6, the set of digits 5, 6, 7) have three consecutive digits. So A alone leaves two candidates, 324 and 576, and cannot fix N by itself.
Step 3: Check statement B alone.
Statement B says N is divisible by 18. From the list, 144, 324, 576 and 900 are all divisible by 18. That is four candidates, so B alone is not enough either.
Step 4: Check statement C alone.
Statement C says the units digit of \(N^2\) equals c, the units digit of N. This is true for several of the 11 numbers, so C alone also fails to pin down one value.
Step 5: Combine A and C.
A narrows N to 324 or 576. Check clue C on these two: \(324^2 = 104976\), units digit 6, but c = 4 for 324, so 324 fails. \(576^2 = 331776\), units digit 6, and c = 6 for 576, so 576 passes. A and C together fix N = 576, so this pair is sufficient.
Step 6: Combine B and C.
B narrows N to 144, 324, 576 or 900. Checking clue C the same way, N = 576 satisfies both the divisibility by 18 and the units digit condition together, so B and C also settle on N = 576. This pair is sufficient too.
Step 7: Rule out the other options.
A and B together only narrow N to 324 or 576, so A and B alone do not fix N, which rules out choice (1). Since B and C alone are also sufficient, choice (2), which claims only A and C work, is too narrow. Because BOTH the pair A, C and the pair B, C independently pin down N = 576, the answer must allow either pairing.
Final Answer:
Either A and C together, or B and C together, are each enough to fix N = 576. \[ \boxed{\text{Either A and C together, or B and C together, are sufficient}} \]
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