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

Hint:

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.

Answer 1

The Correct Option is D

Step 1: Translate "only two factors excluding 1 and itself" into a divisor count.
Every number has 1 and itself as factors. If a four-digit number 'abcd' has exactly two more factors besides these, its total number of factors is 2 + 2 = 4.
A number has exactly 4 factors only in two shapes: it is the cube of a prime (\(p^3\)), or it is the product of two distinct primes (\(p \times q\)).

Step 2: Check the digit conditions for each option.
The first two digits (ab) must form a perfect square, and the last two digits (cd) must be one more than a perfect square.
(a) 1626: ab = 16 (perfect square, yes), cd = 26 = 25 + 1 (yes, since 25 is 5 squared).
(b) 1665: ab = 16 (yes), cd = 65 = 64 + 1 (yes, since 64 is 8 squared).
(c) 2565: ab = 25 (yes), cd = 65 = 64 + 1 (yes).
(d) 2582: ab = 25 (yes), cd = 82 = 81 + 1 (yes, since 81 is 9 squared).
All four options pass the digit check, so we must use the factor count to break the tie.

Step 3: Factorise each option and count the total factors.
1626 = 2 x 3 x 271, three distinct primes, so it has \( 2 \times 2 \times 2 = 8 \) factors. Too many.
1665 = \(3^2 \times 5 \times 37\), so it has \( 3 \times 2 \times 2 = 12 \) factors. Too many.
2565 = \(3^3 \times 5 \times 19\), so it has \( 4 \times 2 \times 2 = 16 \) factors. Too many.
2582 = 2 x 1291, and 1291 checks out as a prime number, since no prime up to its square root, about 35.9, divides it. So 2582 is a product of exactly two distinct primes, giving \( 2 \times 2 = 4 \) factors in total.

Step 4: Confirm the answer.
Only 2582 has exactly 4 total factors, namely 1, 2, 1291 and 2582, which means exactly two factors besides 1 and itself, matching the condition in the question.

Final Answer:
The number is 2582. \[ \boxed{2582} \]