# 1288 Mystery Puzzle

How difficult is this mystery level puzzle? That is part of the mystery, but I assure you that if you use logic and basic multiplication facts you can find the unique solution. Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

That was puzzle #1288. Now I’ll tell you some facts about the number 1288.

• 1288 is a composite number.
• Prime factorization: 1288 = 2 × 2 × 2 × 7 × 23, which can be written 1288 = 2³ × 7 × 23
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1288 has exactly 16 factors.
• Factors of 1288: 1, 2, 4, 7, 8, 14, 23, 28, 46, 56, 92, 161, 184, 322, 644, 1288
• Factor pairs: 1288 = 1 × 1288, 2 × 644, 4 × 322, 7 × 184, 8 × 161, 14 × 92, 23 × 56, or 28 × 46
• Taking the factor pair with the largest square number factor, we get √1288 = (√4)(√322) = 2√322 ≈ 37.88872 1288 has four factor pairs that contain only even factors so 1288 can be written as the difference of two squares four different ways:
323² – 321² = 1288
163² – 159² = 1288
53² – 39² = 1288
37² – 9² = 1288