AMC 8 Daily Practice - Multiplication Formula

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Problem 8 Medium

A four-digit perfect square has all its digits less than 7. When 3 is added to each of its digits, the resulting number is also a perfect square. What is the original four-digit number?

  • A.

    1089

  • B.

    1156

  • C.

    1225

  • D.

    1296

  • E.

    1369

Answer:B

Let the number be \overline{abcd} = m^2.

After adding 3 to each digit: \overline{(a+3)(b+3)(c+3)(d+3)} = n^2.

n^2 - m^2 = 3333 = (n-m)(n+m).

Factor pairs of 3333: 1 \times 3333, 3 \times 1111, 11 \times 303, 33 \times 101. Only 33 \times 101 satisfies.

n+m=101 and n-m=33 \implies m=34.

Original number: 34^2 = 1156.

Final result: \boxed{1156}

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