# 1328 Christmas Tree

Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Now I’ll share some facts about the puzzle number, 1328:

• 1328 is a composite number.
• Prime factorization: 1328 = 2 × 2 × 2 × 2 × 83, which can be written 1328 = 2⁴ × 83
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1328 has exactly 10 factors.
• Factors of 1328: 1, 2, 4, 8, 16, 83, 166, 332, 664, 1328
• Factor pairs: 1328 = 1 × 1328, 2 × 664, 4 × 332, 8 × 166, or 16 × 83
• Taking the factor pair with the largest square number factor, we get √1328 = (√16)(√83) = 4√83 ≈ 36.44173

Because 28 is divisible by 4, but not by 8, and 3 (the digit before the 28) is an odd number, I know that 1328 is divisible by 8. I can use that fact to make this simple factor tree:

1328 is the difference of two squares three different ways:
333² – 331² = 1328
168² – 164² = 1328
87² – 79²  = 1328