Question:hard

'abcd' is a four-digit number. It has only two factors excluding 1 and itself. In addition, the first two digits form a perfect square and the next two digits form a number which is one more than a perfect square. Which of the following could be the number?

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A number with exactly two factors besides 1 and itself has 4 factors in total, so it is either the cube of a prime or the product of two distinct primes. Factorise each option to check.
Updated On: Jul 21, 2026
  • 1626
  • 1665
  • 2565
  • 2582
Show Solution

The Correct Option is D

Solution and Explanation

This question is easiest to crack by working backward from the factor condition, since that is the strictest test. A number with "only two factors excluding 1 and itself" has exactly 4 factors in total, which only happens when the number is a cube of a single prime or the product of two distinct primes. Let's factorise every option first and see which ones survive.

  1. 1626: This splits as 2 x 3 x 271, three different prime factors, giving 8 factors in total (1, 2, 3, 6, 271, 542, 813, 1626). That is too many, so 1626 is ruled out.
  2. 1665: This splits as \(3^2 \times 5 \times 37\), giving \(3 \times 2 \times 2 = 12\) factors in total. Also too many, so 1665 is ruled out.
  3. 2565: This splits as \(3^3 \times 5 \times 19\), giving \(4 \times 2 \times 2 = 16\) factors in total. Ruled out for the same reason.
  4. 2582: This splits as 2 x 1291. Testing 1291 against every prime up to its square root, about 36, shows none of them divide it, so 1291 is prime. That makes 2582 a product of exactly two distinct primes, with exactly 4 factors in total: 1, 2, 1291, and 2582.

Only 2582 survives the factor-count test. As a final check, its digits also fit the other two conditions: the first two digits, 25, form a perfect square, 5 squared, and the last two digits, 82, are one more than 81, which is 9 squared.

Let's summarize:

  • A number with exactly two factors besides 1 and itself must have 4 factors in total.
  • Factorising all four options shows only 2582 has exactly 4 factors.
  • 2582 also fits the perfect-square and one-more-than-a-perfect-square digit pattern.

So the number is 2582.

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