# 768 and Level 1

768 is made from three consecutive numbers so it is divisible by 3.

• 768 is a composite number.
• Prime factorization: 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3, which can be written 768 = 2⁸ × 3
• The exponents in the prime factorization are 8 and 1. Adding one to each and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 768 has exactly 18 factors.
• Factors of 768: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768
• Factor pairs: 768 = 1 × 768, 2 × 384, 3 × 256, 4 × 192, 6 × 128, 8 × 96, 12 × 64, 16 × 48 or 24 × 32
• Taking the factor pair with the largest square number factor, we get √768 = (√256)(√3) = 16√3 ≈ 27.7128129. Today’s puzzle is relatively easy: Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Here’s a few more facts about the number 768:

255 + 256 + 257 = 768. That is the ONLY way to write 768 as the sum of consecutive positive whole numbers.

There is also only one way to write 768 as the sum of three square numbers:

• 16² + 16² + 16² = 768.

768 is the sum of eight consecutive prime numbers: 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 = 768.

768 is 1100000000 in BASE 2, 30000 in BASE 4, and 300 in BASE 16.

768 is also a palindrome in a few bases:

• 363 BASE 15; note that 3(225) + 6(15) + 3(1) = 768
• OO BASE 31 (O = 24 base 10); note that 24(31) + 24(1) = 768
• CC BASE 63 (C = 12 base 10); note that 12(63) + 12(1) = 768
• 66 BASE 127; note that 6(127) + 6(1) = 768
• 33 BASE 255; note that 3(255) + 3(1) = 768

What do all those BASES have in common? They are all one less than a power of 2.

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