Question

'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?

• A. 1626
• B. 1665
• C. 2565
• D. 2582
• E. 3682

Hint:

The number must have exactly 4 factors in total (so it is \(p \times q\) or \(p^3\)); check the digit-square condition first to shortlist, then factorise.

Answer 1

The Correct Option is D

Step 1: Decode the condition on factors.
If 'abcd' has only two factors excluding 1 and itself, the number has exactly 4 factors in total. A number has exactly 4 factors only when it is of the form \(p^3\) (p prime) or \(p \times q\) with p, q distinct primes.

Step 2: Decode the digit condition.
The first two digits (ab) must be a perfect square, and the last two digits (cd) must be one more than a perfect square, i.e. \(cd = k^2+1\).

Step 3: Filter the options on the digit condition first.
(a) 1626: ab = 16 = \(4^2\) ✓, cd = 26 = \(5^2+1\) ✓
(b) 1665: ab = 16 ✓, cd = 65 = \(8^2+1\) ✓
(c) 2565: ab = 25 = \(5^2\) ✓, cd = 65 = \(8^2+1\) ✓
(d) 2582: ab = 25 ✓, cd = 82 = \(9^2+1\) ✓
(e) 3682: ab = 36 = \(6^2\) ✓, cd = 82 ✓
All five pass the digit test, so the factor condition must decide it.

Step 4: Factorise every option.
1626 = \(2 \times 3 \times 271\) → 8 factors, rejected.
1665 = \(3^2 \times 5 \times 37\) → 12 factors, rejected.
2565 = \(3^3 \times 5 \times 19\) → 16 factors, rejected.
2582 = \(2 \times 1291\), and 1291 has no divisor among primes up to \(\sqrt{1291} \approx 35.9\) (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 all fail), so 1291 is prime. 2582 has exactly 4 factors: 1, 2, 1291, 2582 → excluding 1 and itself gives exactly two factors (2 and 1291). This satisfies the condition.
3682 = \(2 \times 7 \times 263\) → 8 factors, rejected.

Step 5: Conclusion.
Only 2582 satisfies both conditions.